| L(s) = 1 | − 4-s + 4·5-s + 2·9-s + 4·11-s + 16-s − 4·20-s + 11·25-s + 4·29-s + 8·31-s − 2·36-s − 4·44-s + 8·45-s − 49-s + 16·55-s − 8·59-s + 4·61-s − 64-s − 4·71-s + 4·80-s − 5·81-s − 8·89-s + 8·99-s − 11·100-s − 12·101-s + 12·109-s − 4·116-s − 10·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.78·5-s + 2/3·9-s + 1.20·11-s + 1/4·16-s − 0.894·20-s + 11/5·25-s + 0.742·29-s + 1.43·31-s − 1/3·36-s − 0.603·44-s + 1.19·45-s − 1/7·49-s + 2.15·55-s − 1.04·59-s + 0.512·61-s − 1/8·64-s − 0.474·71-s + 0.447·80-s − 5/9·81-s − 0.847·89-s + 0.804·99-s − 1.09·100-s − 1.19·101-s + 1.14·109-s − 0.371·116-s − 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.333518884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.333518884\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−10.01473249054065170531924362528, −9.947651233193620163164308221581, −9.557191311076260028206441785700, −9.031026619630880856163774547220, −8.911735968218787416251025276623, −8.260614212329097415404659968334, −7.936240058426380646585504687125, −7.11552620636637068741016097092, −6.79357310902524279559231075030, −6.46259800411057506655822018909, −5.97306055380208959316859735337, −5.60966200230250760237120124227, −5.04485606583111317638498245462, −4.38120949634964090312893319468, −4.35186457097699174622434125866, −3.38273566707552092115858371419, −2.86831250698150516139532963275, −2.16220496185843469743226497659, −1.48483961568717874790380008330, −1.00399255309798036665391335563,
1.00399255309798036665391335563, 1.48483961568717874790380008330, 2.16220496185843469743226497659, 2.86831250698150516139532963275, 3.38273566707552092115858371419, 4.35186457097699174622434125866, 4.38120949634964090312893319468, 5.04485606583111317638498245462, 5.60966200230250760237120124227, 5.97306055380208959316859735337, 6.46259800411057506655822018909, 6.79357310902524279559231075030, 7.11552620636637068741016097092, 7.936240058426380646585504687125, 8.260614212329097415404659968334, 8.911735968218787416251025276623, 9.031026619630880856163774547220, 9.557191311076260028206441785700, 9.947651233193620163164308221581, 10.01473249054065170531924362528
