| L(s) = 1 | + 31.5·2-s − 81·3-s + 483.·4-s + 1.95e3·5-s − 2.55e3·6-s − 4.90e3·7-s − 888.·8-s + 6.56e3·9-s + 6.17e4·10-s − 7.90e4·11-s − 3.91e4·12-s − 1.84e3·13-s − 1.54e5·14-s − 1.58e5·15-s − 2.75e5·16-s + 4.59e5·17-s + 2.07e5·18-s + 1.08e6·19-s + 9.46e5·20-s + 3.97e5·21-s − 2.49e6·22-s + 2.20e6·23-s + 7.20e4·24-s + 1.87e6·25-s − 5.82e4·26-s − 5.31e5·27-s − 2.37e6·28-s + ⋯ |
| L(s) = 1 | + 1.39·2-s − 0.577·3-s + 0.944·4-s + 1.39·5-s − 0.805·6-s − 0.771·7-s − 0.0767·8-s + 0.333·9-s + 1.95·10-s − 1.62·11-s − 0.545·12-s − 0.0179·13-s − 1.07·14-s − 0.808·15-s − 1.05·16-s + 1.33·17-s + 0.464·18-s + 1.90·19-s + 1.32·20-s + 0.445·21-s − 2.27·22-s + 1.64·23-s + 0.0442·24-s + 0.959·25-s − 0.0250·26-s − 0.192·27-s − 0.729·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.436602954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.436602954\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 - 31.5T + 512T^{2} \) |
| 5 | \( 1 - 1.95e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.90e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.90e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.84e3T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.59e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.08e6T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.20e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.56e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.82e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.05e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.66e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.40e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.70e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.35e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 1.59e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.14e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.06e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.12e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.40e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 9.07e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.56e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 3.71e8T + 7.60e17T^{2} \) |
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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
−11.21363268615545266139964628143, −10.04115360703614605588320520381, −9.445044210550205354662061085262, −7.51267438579237762228247911325, −6.33000172879414084355269900458, −5.37379265227051123769857395359, −5.18776386131649470197859621603, −3.31112268784197654499451265893, −2.54698173657369464280833562063, −0.871960846110275261303596227028,
0.871960846110275261303596227028, 2.54698173657369464280833562063, 3.31112268784197654499451265893, 5.18776386131649470197859621603, 5.37379265227051123769857395359, 6.33000172879414084355269900458, 7.51267438579237762228247911325, 9.445044210550205354662061085262, 10.04115360703614605588320520381, 11.21363268615545266139964628143
