| L(s) = 1 | − 37.2·2-s + 6.56e3·3-s − 1.29e5·4-s − 2.44e5·6-s − 4.45e6·7-s + 9.71e6·8-s + 4.30e7·9-s + 8.94e8·11-s − 8.50e8·12-s + 4.68e9·13-s + 1.66e8·14-s + 1.66e10·16-s − 7.14e9·17-s − 1.60e9·18-s + 1.60e10·19-s − 2.92e10·21-s − 3.33e10·22-s − 2.00e11·23-s + 6.37e10·24-s − 1.74e11·26-s + 2.82e11·27-s + 5.77e11·28-s − 3.38e12·29-s − 7.72e11·31-s − 1.89e12·32-s + 5.86e12·33-s + 2.66e11·34-s + ⋯ |
| L(s) = 1 | − 0.102·2-s + 0.577·3-s − 0.989·4-s − 0.0594·6-s − 0.292·7-s + 0.204·8-s + 0.333·9-s + 1.25·11-s − 0.571·12-s + 1.59·13-s + 0.0300·14-s + 0.968·16-s − 0.248·17-s − 0.0343·18-s + 0.216·19-s − 0.168·21-s − 0.129·22-s − 0.534·23-s + 0.118·24-s − 0.163·26-s + 0.192·27-s + 0.288·28-s − 1.25·29-s − 0.162·31-s − 0.304·32-s + 0.726·33-s + 0.0255·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(2.300645389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.300645389\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 6.56e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 37.2T + 1.31e5T^{2} \) |
| 7 | \( 1 + 4.45e6T + 2.32e14T^{2} \) |
| 11 | \( 1 - 8.94e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 4.68e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 7.14e9T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.60e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 2.00e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 3.38e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 7.72e11T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.98e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 6.17e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 6.28e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.59e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 4.53e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 7.87e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 8.05e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 4.65e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 9.90e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.59e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.37e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 7.48e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + 2.79e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.10e17T + 5.95e33T^{2} \) |
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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
−11.12740954695821547728155007871, −9.802420690496410550480153491774, −8.977533743606171727443612131221, −8.269469691520519407455509377053, −6.80837839677707112515141351831, −5.57273463905846721790358600697, −4.00633430203551714703919111117, −3.56040711718281874464918277227, −1.73982695776437796873710705938, −0.72050077441615381218009873792,
0.72050077441615381218009873792, 1.73982695776437796873710705938, 3.56040711718281874464918277227, 4.00633430203551714703919111117, 5.57273463905846721790358600697, 6.80837839677707112515141351831, 8.269469691520519407455509377053, 8.977533743606171727443612131221, 9.802420690496410550480153491774, 11.12740954695821547728155007871
