| L(s) = 1 | − 11.1·2-s + 27·3-s − 3.03·4-s + 497.·5-s − 301.·6-s + 1.04e3·7-s + 1.46e3·8-s + 729·9-s − 5.55e3·10-s − 8.36e3·11-s − 81.8·12-s − 1.08e4·13-s − 1.17e4·14-s + 1.34e4·15-s − 1.59e4·16-s + 3.41e3·17-s − 8.14e3·18-s + 1.24e4·19-s − 1.50e3·20-s + 2.82e4·21-s + 9.35e4·22-s − 3.19e3·23-s + 3.95e4·24-s + 1.69e5·25-s + 1.21e5·26-s + 1.96e4·27-s − 3.17e3·28-s + ⋯ |
| L(s) = 1 | − 0.988·2-s + 0.577·3-s − 0.0236·4-s + 1.77·5-s − 0.570·6-s + 1.15·7-s + 1.01·8-s + 0.333·9-s − 1.75·10-s − 1.89·11-s − 0.0136·12-s − 1.36·13-s − 1.14·14-s + 1.02·15-s − 0.975·16-s + 0.168·17-s − 0.329·18-s + 0.417·19-s − 0.0421·20-s + 0.666·21-s + 1.87·22-s − 0.0547·23-s + 0.583·24-s + 2.16·25-s + 1.35·26-s + 0.192·27-s − 0.0273·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.189545430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.189545430\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 103 | \( 1 - 1.09e6T \) |
| good | 2 | \( 1 + 11.1T + 128T^{2} \) |
| 5 | \( 1 - 497.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.04e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 8.36e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.08e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.41e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.24e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.19e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.95e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.61e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.01e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.68e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.71e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.94e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.43e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.18e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.69e4T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.29e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.55e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.91e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.02e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.01e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.70e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.02e6T + 8.07e13T^{2} \) |
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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
−9.954889375056204935219120805701, −9.862291066565182071967513789430, −8.516518321737357957361250601458, −8.004787929057011531637244044421, −6.95030739132039870478386067583, −5.18922784767532841641880162257, −4.94619752582227122862798028843, −2.58070326619576007390997073315, −2.01408802011074230220294819640, −0.829331670527037989120752079917,
0.829331670527037989120752079917, 2.01408802011074230220294819640, 2.58070326619576007390997073315, 4.94619752582227122862798028843, 5.18922784767532841641880162257, 6.95030739132039870478386067583, 8.004787929057011531637244044421, 8.516518321737357957361250601458, 9.862291066565182071967513789430, 9.954889375056204935219120805701
