| L(s) = 1 | − 16·2-s + 159.·3-s + 256·4-s + 1.76e3·5-s − 2.55e3·6-s + 6.24e3·7-s − 4.09e3·8-s + 5.76e3·9-s − 2.82e4·10-s − 4.10e4·11-s + 4.08e4·12-s + 2.85e4·13-s − 9.98e4·14-s + 2.81e5·15-s + 6.55e4·16-s + 9.82e4·17-s − 9.21e4·18-s + 5.07e5·19-s + 4.52e5·20-s + 9.95e5·21-s + 6.57e5·22-s + 1.89e6·23-s − 6.53e5·24-s + 1.17e6·25-s − 4.56e5·26-s − 2.22e6·27-s + 1.59e6·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.13·3-s + 0.5·4-s + 1.26·5-s − 0.803·6-s + 0.982·7-s − 0.353·8-s + 0.292·9-s − 0.894·10-s − 0.845·11-s + 0.568·12-s + 0.277·13-s − 0.694·14-s + 1.43·15-s + 0.250·16-s + 0.285·17-s − 0.206·18-s + 0.892·19-s + 0.632·20-s + 1.11·21-s + 0.598·22-s + 1.41·23-s − 0.401·24-s + 0.599·25-s − 0.196·26-s − 0.804·27-s + 0.491·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.464925324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.464925324\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 3 | \( 1 - 159.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.76e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 6.24e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.10e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 9.82e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.07e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.89e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.21e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.09e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.77e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.79e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.22e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.23e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.92e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.72e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.36e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.21e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.66e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.13e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.79e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.50e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.04e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.21e9T + 7.60e17T^{2} \) |
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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
−15.18073426154400514394683237250, −14.18396720879773502315784693174, −13.17800317553637696510245991683, −11.12931914039912323815731995571, −9.763188566684373617740618452605, −8.726340941141516650853413775151, −7.54660710920621517861496736662, −5.47438258662496058177250840780, −2.79540920494524299639557192985, −1.53739968747364262683868535690,
1.53739968747364262683868535690, 2.79540920494524299639557192985, 5.47438258662496058177250840780, 7.54660710920621517861496736662, 8.726340941141516650853413775151, 9.763188566684373617740618452605, 11.12931914039912323815731995571, 13.17800317553637696510245991683, 14.18396720879773502315784693174, 15.18073426154400514394683237250
