| L(s) = 1 | + 59·4-s − 72·9-s + 792·11-s + 1.71e3·16-s + 6.53e3·19-s − 1.33e4·29-s − 40·31-s − 4.24e3·36-s − 1.20e4·41-s + 4.67e4·44-s − 4.80e3·49-s + 8.78e4·59-s − 1.29e5·61-s + 5.76e4·64-s + 1.94e5·71-s + 3.85e5·76-s − 1.02e5·79-s − 9.57e4·81-s − 1.68e5·89-s − 5.70e4·99-s − 3.11e5·101-s + 3.94e5·109-s − 7.90e5·116-s + 1.86e5·121-s − 2.36e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.84·4-s − 0.296·9-s + 1.97·11-s + 1.67·16-s + 4.15·19-s − 2.95·29-s − 0.00747·31-s − 0.546·36-s − 1.12·41-s + 3.63·44-s − 2/7·49-s + 3.28·59-s − 4.45·61-s + 1.75·64-s + 4.58·71-s + 7.65·76-s − 1.84·79-s − 1.62·81-s − 2.25·89-s − 0.584·99-s − 3.03·101-s + 3.18·109-s − 5.45·116-s + 1.15·121-s − 0.0137·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.481122574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.481122574\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 59 T^{2} + 441 p^{2} T^{4} - 59 p^{10} T^{6} + p^{20} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 8 p^{2} T^{2} + 1246 p^{4} T^{4} + 8 p^{12} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 36 p T + 142198 T^{2} - 36 p^{6} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 971456 T^{2} + 483935131182 T^{4} - 971456 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 3894812 T^{2} + 7557700009734 T^{4} - 3894812 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 3266 T + 7614270 T^{2} - 3266 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 14370716 T^{2} + 114438939827814 T^{4} - 14370716 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 6696 T + 51326470 T^{2} + 6696 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 20 T + 53103102 T^{2} + 20 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 249646316 T^{2} + 25036606707861462 T^{4} - 249646316 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 6048 T + 223864366 T^{2} + 6048 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 69887332 T^{2} + 41464568025667254 T^{4} + 69887332 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 448581164 T^{2} + 128102309424078822 T^{4} - 448581164 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1626727436 T^{2} + 1011280212489797334 T^{4} - 1626727436 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 43938 T + 1852599934 T^{2} - 43938 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 64754 T + 2408321418 T^{2} + 64754 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4985671724 T^{2} + 9826827410456194614 T^{4} - 4985671724 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 97416 T + 5729557966 T^{2} - 97416 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 7352672444 T^{2} + 21990800559226590630 T^{4} - 7352672444 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 51256 T + 3645565854 T^{2} + 51256 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1776439640 T^{2} - 17030759318944523202 T^{4} - 1776439640 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 84276 T + 5915697430 T^{2} + 84276 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 31916456540 T^{2} + \)\(40\!\cdots\!98\)\( T^{4} - 31916456540 p^{10} T^{6} + p^{20} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.471920739833752295394099375015, −7.73358857736965973562786077151, −7.69027436718918716209788719979, −7.63834038668581608499324708238, −7.25308632207355758057379669385, −6.93467580918626380236630044543, −6.68838416238483559044529495390, −6.61659349724714654412557693828, −6.18774542718380142226486971078, −5.79060854992652268465546655804, −5.38155509493841973191704957488, −5.32542772183890970907090379396, −5.29522454686024614241209559123, −4.50489662729762552978702557329, −3.98023680574650254558146979086, −3.73217024238547272792880930843, −3.55482997455285191228931430558, −3.02417088929904646044511933564, −2.98495650882085153740970856196, −2.33162275356981750289669815764, −2.02188499118541663053174572586, −1.37031054122088129466497359234, −1.31562820103702237273734432952, −1.11996315233722222587323019420, −0.26095649717168051161530436504,
0.26095649717168051161530436504, 1.11996315233722222587323019420, 1.31562820103702237273734432952, 1.37031054122088129466497359234, 2.02188499118541663053174572586, 2.33162275356981750289669815764, 2.98495650882085153740970856196, 3.02417088929904646044511933564, 3.55482997455285191228931430558, 3.73217024238547272792880930843, 3.98023680574650254558146979086, 4.50489662729762552978702557329, 5.29522454686024614241209559123, 5.32542772183890970907090379396, 5.38155509493841973191704957488, 5.79060854992652268465546655804, 6.18774542718380142226486971078, 6.61659349724714654412557693828, 6.68838416238483559044529495390, 6.93467580918626380236630044543, 7.25308632207355758057379669385, 7.63834038668581608499324708238, 7.69027436718918716209788719979, 7.73358857736965973562786077151, 8.471920739833752295394099375015
