| L(s) = 1 | − 51.1·2-s + 243·3-s + 564.·4-s − 3.12e3·5-s − 1.24e4·6-s + 4.57e4·7-s + 7.58e4·8-s + 5.90e4·9-s + 1.59e5·10-s − 5.97e5·11-s + 1.37e5·12-s − 9.90e5·13-s − 2.33e6·14-s − 7.59e5·15-s − 5.03e6·16-s − 6.61e6·17-s − 3.01e6·18-s + 5.76e5·19-s − 1.76e6·20-s + 1.11e7·21-s + 3.05e7·22-s − 4.64e7·23-s + 1.84e7·24-s + 9.76e6·25-s + 5.06e7·26-s + 1.43e7·27-s + 2.58e7·28-s + ⋯ |
| L(s) = 1 | − 1.12·2-s + 0.577·3-s + 0.275·4-s − 0.447·5-s − 0.652·6-s + 1.02·7-s + 0.818·8-s + 0.333·9-s + 0.505·10-s − 1.11·11-s + 0.159·12-s − 0.739·13-s − 1.16·14-s − 0.258·15-s − 1.19·16-s − 1.12·17-s − 0.376·18-s + 0.0534·19-s − 0.123·20-s + 0.594·21-s + 1.26·22-s − 1.50·23-s + 0.472·24-s + 0.199·25-s + 0.835·26-s + 0.192·27-s + 0.283·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 5 | \( 1 + 3.12e3T \) |
| good | 2 | \( 1 + 51.1T + 2.04e3T^{2} \) |
| 7 | \( 1 - 4.57e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 5.97e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 9.90e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 6.61e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 5.76e5T + 1.16e14T^{2} \) |
| 23 | \( 1 + 4.64e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.59e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.04e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.80e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 6.63e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.76e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.39e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 2.28e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 3.12e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 6.71e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 5.93e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.13e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 9.39e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.59e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.01e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.46e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + 7.50e10T + 7.15e21T^{2} \) |
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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
−16.12541011382593705086950241925, −14.77355892743574205174971100831, −13.27845858293958475735421338763, −11.28409358505381805823128715415, −9.898947466332476983520570529108, −8.370457475004131918886644025987, −7.59113257631661872386870158637, −4.59869489982995921707703275522, −2.03380616471124539964325942731, 0,
2.03380616471124539964325942731, 4.59869489982995921707703275522, 7.59113257631661872386870158637, 8.370457475004131918886644025987, 9.898947466332476983520570529108, 11.28409358505381805823128715415, 13.27845858293958475735421338763, 14.77355892743574205174971100831, 16.12541011382593705086950241925
