| L(s) = 1 | + 20·5-s − 60·11-s − 24·19-s + 275·25-s − 128·29-s − 624·31-s + 748·41-s + 362·49-s − 1.20e3·55-s + 1.02e3·59-s + 1.50e3·61-s + 1.84e3·71-s − 144·79-s − 580·89-s − 480·95-s − 208·101-s − 1.14e3·109-s + 38·121-s + 3.00e3·125-s + 127-s + 131-s + 137-s + 139-s − 2.56e3·145-s + 149-s + 151-s − 1.24e4·155-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.64·11-s − 0.289·19-s + 11/5·25-s − 0.819·29-s − 3.61·31-s + 2.84·41-s + 1.05·49-s − 2.94·55-s + 2.25·59-s + 3.16·61-s + 3.08·71-s − 0.205·79-s − 0.690·89-s − 0.518·95-s − 0.204·101-s − 1.00·109-s + 0.0285·121-s + 2.14·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 1.46·145-s + 0.000549·149-s + 0.000538·151-s − 6.46·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.705975955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.705975955\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 362 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2950 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4926 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 12 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 19150 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 64 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 312 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 82262 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 374 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 60010 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 190222 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 98838 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 510 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 754 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 454070 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 924 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 662434 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1119238 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 290 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1683970 T^{2} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.541742100117204704659214908722, −9.139510755152034486241557464282, −8.436960427712506281639273600523, −8.419758107668550265432740129874, −7.60274664120605105313852704791, −7.36052496136214643269450979987, −6.96264074401133518583514882226, −6.50482752304830160907328819271, −5.88574156159852084089795592708, −5.44376429563268052804529886726, −5.37802717042391699557181664807, −5.20915532550212545101821080641, −4.12519231879542606167585001214, −3.92669585153679174060811840533, −3.20567447041666559858191237600, −2.48062156862639189626744594477, −2.14735511111703054915887147893, −2.03941806773108514571708023956, −1.01241873053732260153419952593, −0.46209766901744422265592530698,
0.46209766901744422265592530698, 1.01241873053732260153419952593, 2.03941806773108514571708023956, 2.14735511111703054915887147893, 2.48062156862639189626744594477, 3.20567447041666559858191237600, 3.92669585153679174060811840533, 4.12519231879542606167585001214, 5.20915532550212545101821080641, 5.37802717042391699557181664807, 5.44376429563268052804529886726, 5.88574156159852084089795592708, 6.50482752304830160907328819271, 6.96264074401133518583514882226, 7.36052496136214643269450979987, 7.60274664120605105313852704791, 8.419758107668550265432740129874, 8.436960427712506281639273600523, 9.139510755152034486241557464282, 9.541742100117204704659214908722
