| L(s) = 1 | + 4·2-s − 10.0·3-s + 16·4-s + 80.2·5-s − 40.2·6-s + 208.·7-s + 64·8-s − 141.·9-s + 320.·10-s − 459.·11-s − 161.·12-s − 169·13-s + 834.·14-s − 807.·15-s + 256·16-s + 1.78e3·17-s − 566.·18-s − 2.12e3·19-s + 1.28e3·20-s − 2.10e3·21-s − 1.83e3·22-s − 3.09e3·23-s − 644.·24-s + 3.30e3·25-s − 676·26-s + 3.87e3·27-s + 3.33e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.645·3-s + 0.5·4-s + 1.43·5-s − 0.456·6-s + 1.60·7-s + 0.353·8-s − 0.582·9-s + 1.01·10-s − 1.14·11-s − 0.322·12-s − 0.277·13-s + 1.13·14-s − 0.926·15-s + 0.250·16-s + 1.49·17-s − 0.412·18-s − 1.34·19-s + 0.717·20-s − 1.03·21-s − 0.809·22-s − 1.22·23-s − 0.228·24-s + 1.05·25-s − 0.196·26-s + 1.02·27-s + 0.804·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.223178590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.223178590\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 13 | \( 1 + 169T \) |
| good | 3 | \( 1 + 10.0T + 243T^{2} \) |
| 5 | \( 1 - 80.2T + 3.12e3T^{2} \) |
| 7 | \( 1 - 208.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 459.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.78e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.12e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.09e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.89e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.10e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.32e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 913.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.22e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.84e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.39e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.83e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.25e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.21e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.85e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.13e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.54e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.35e5T + 8.58e9T^{2} \) |
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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
−16.65393141509281787789022351935, −14.75021561878678320725060837236, −14.05199997379584470331190154786, −12.67548542762211480037290975431, −11.28212664178934284505791888180, −10.22118613187508597569152732301, −8.041958211122648473035761903482, −5.87771978695044039499789698161, −5.09509794415253796088132261534, −2.06394894328285352622557765711,
2.06394894328285352622557765711, 5.09509794415253796088132261534, 5.87771978695044039499789698161, 8.041958211122648473035761903482, 10.22118613187508597569152732301, 11.28212664178934284505791888180, 12.67548542762211480037290975431, 14.05199997379584470331190154786, 14.75021561878678320725060837236, 16.65393141509281787789022351935
