| L(s) = 1 | + 26.5·2-s + 204.·3-s + 194.·4-s + 2.37e3·5-s + 5.44e3·6-s − 6.24e3·7-s − 8.43e3·8-s + 2.23e4·9-s + 6.30e4·10-s + 8.26e4·11-s + 3.98e4·12-s + 1.10e5·13-s − 1.65e5·14-s + 4.86e5·15-s − 3.23e5·16-s + 5.93e5·18-s + 5.62e5·19-s + 4.61e5·20-s − 1.27e6·21-s + 2.19e6·22-s + 1.19e6·23-s − 1.72e6·24-s + 3.67e6·25-s + 2.94e6·26-s + 5.40e5·27-s − 1.21e6·28-s − 1.98e6·29-s + ⋯ |
| L(s) = 1 | + 1.17·2-s + 1.46·3-s + 0.379·4-s + 1.69·5-s + 1.71·6-s − 0.982·7-s − 0.728·8-s + 1.13·9-s + 1.99·10-s + 1.70·11-s + 0.555·12-s + 1.07·13-s − 1.15·14-s + 2.47·15-s − 1.23·16-s + 1.33·18-s + 0.989·19-s + 0.644·20-s − 1.43·21-s + 1.99·22-s + 0.890·23-s − 1.06·24-s + 1.88·25-s + 1.26·26-s + 0.195·27-s − 0.373·28-s − 0.521·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(10.24835955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.24835955\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 26.5T + 512T^{2} \) |
| 3 | \( 1 - 204.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.37e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 6.24e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.26e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.10e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 5.62e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.19e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.98e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 9.43e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.84e5T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.19e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.42e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.11e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.19e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 4.21e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.01e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.11e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.37e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.34e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.77e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.29e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.82e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.77e7T + 7.60e17T^{2} \) |
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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
−9.738352454868312531248961445312, −9.260775795736082577394225998683, −8.807502581193074046177647415671, −6.93428671394015516266102792898, −6.21507417930008078462622322553, −5.32983908334342410838320471479, −3.72370150362477269504062957738, −3.36611064418664448727156497269, −2.24344424293483917920573533788, −1.24425412865183904802723062683,
1.24425412865183904802723062683, 2.24344424293483917920573533788, 3.36611064418664448727156497269, 3.72370150362477269504062957738, 5.32983908334342410838320471479, 6.21507417930008078462622322553, 6.93428671394015516266102792898, 8.807502581193074046177647415671, 9.260775795736082577394225998683, 9.738352454868312531248961445312
