| L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s + 4·11-s − 13-s − 15-s − 17-s − 8·19-s + 4·21-s − 4·23-s + 25-s + 27-s − 2·29-s + 4·33-s − 4·35-s − 6·37-s − 39-s − 6·41-s − 4·43-s − 45-s + 9·49-s − 51-s + 2·53-s − 4·55-s − 8·57-s + 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s − 0.242·17-s − 1.83·19-s + 0.872·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.696·33-s − 0.676·35-s − 0.986·37-s − 0.160·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s + 9/7·49-s − 0.140·51-s + 0.274·53-s − 0.539·55-s − 1.05·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.545686279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.545686279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.98108698752408, −12.43439140262196, −12.08236729545377, −11.53550187951045, −11.30525675397428, −10.60996164649457, −10.32782535441723, −9.704965637599578, −8.986835767529955, −8.683754864074206, −8.384787484705751, −7.853536180773305, −7.398811849886833, −6.790470316763054, −6.431637480858269, −5.749910154169227, −5.043577251097758, −4.609391574757585, −4.051745123681175, −3.866120552069655, −3.044075099530206, −2.283585110301687, −1.703847195862972, −1.518977555927854, −0.4052119317290674,
0.4052119317290674, 1.518977555927854, 1.703847195862972, 2.283585110301687, 3.044075099530206, 3.866120552069655, 4.051745123681175, 4.609391574757585, 5.043577251097758, 5.749910154169227, 6.431637480858269, 6.790470316763054, 7.398811849886833, 7.853536180773305, 8.384787484705751, 8.683754864074206, 8.986835767529955, 9.704965637599578, 10.32782535441723, 10.60996164649457, 11.30525675397428, 11.53550187951045, 12.08236729545377, 12.43439140262196, 12.98108698752408
