| L(s) = 1 | + 4·3-s − 2·5-s + 7·9-s + 4·11-s − 8·15-s + 8·23-s − 3·25-s + 4·27-s + 16·33-s + 6·37-s − 14·45-s + 6·49-s − 4·53-s − 8·55-s + 12·59-s + 4·67-s + 32·69-s + 24·71-s − 12·75-s − 8·81-s − 26·89-s + 6·97-s + 28·99-s − 16·103-s + 24·111-s − 10·113-s − 16·115-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.894·5-s + 7/3·9-s + 1.20·11-s − 2.06·15-s + 1.66·23-s − 3/5·25-s + 0.769·27-s + 2.78·33-s + 0.986·37-s − 2.08·45-s + 6/7·49-s − 0.549·53-s − 1.07·55-s + 1.56·59-s + 0.488·67-s + 3.85·69-s + 2.84·71-s − 1.38·75-s − 8/9·81-s − 2.75·89-s + 0.609·97-s + 2.81·99-s − 1.57·103-s + 2.27·111-s − 0.940·113-s − 1.49·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.040342453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.040342453\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.527834757835193726747029498818, −8.130905320346842230293799253614, −7.80880047675026696994038925633, −7.38343675003183133018650001814, −6.75675677658737613905670147750, −6.62082714324451009339742326913, −5.66163434916591204950710996791, −5.16707365592353737946128191517, −4.30971861340138374132199477742, −3.95501966037150973513946344449, −3.64483685966352076698862940684, −3.01131321872163071924120162853, −2.60939963678351254594850012675, −1.92256304918733355678144886727, −0.989457762310743048355362377137,
0.989457762310743048355362377137, 1.92256304918733355678144886727, 2.60939963678351254594850012675, 3.01131321872163071924120162853, 3.64483685966352076698862940684, 3.95501966037150973513946344449, 4.30971861340138374132199477742, 5.16707365592353737946128191517, 5.66163434916591204950710996791, 6.62082714324451009339742326913, 6.75675677658737613905670147750, 7.38343675003183133018650001814, 7.80880047675026696994038925633, 8.130905320346842230293799253614, 8.527834757835193726747029498818
