| L(s) = 1 | − 1.27·2-s − 3-s − 0.377·4-s − 1.37·5-s + 1.27·6-s + 0.651·7-s + 3.02·8-s + 9-s + 1.75·10-s + 11-s + 0.377·12-s − 4.27·13-s − 0.829·14-s + 1.37·15-s − 3.10·16-s + 7.33·17-s − 1.27·18-s + 19-s + 0.519·20-s − 0.651·21-s − 1.27·22-s − 1.20·23-s − 3.02·24-s − 3.10·25-s + 5.44·26-s − 27-s − 0.245·28-s + ⋯ |
| L(s) = 1 | − 0.900·2-s − 0.577·3-s − 0.188·4-s − 0.615·5-s + 0.520·6-s + 0.246·7-s + 1.07·8-s + 0.333·9-s + 0.554·10-s + 0.301·11-s + 0.108·12-s − 1.18·13-s − 0.221·14-s + 0.355·15-s − 0.775·16-s + 1.77·17-s − 0.300·18-s + 0.229·19-s + 0.116·20-s − 0.142·21-s − 0.271·22-s − 0.251·23-s − 0.618·24-s − 0.620·25-s + 1.06·26-s − 0.192·27-s − 0.0464·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.27T + 2T^{2} \) |
| 5 | \( 1 + 1.37T + 5T^{2} \) |
| 7 | \( 1 - 0.651T + 7T^{2} \) |
| 13 | \( 1 + 4.27T + 13T^{2} \) |
| 17 | \( 1 - 7.33T + 17T^{2} \) |
| 23 | \( 1 + 1.20T + 23T^{2} \) |
| 29 | \( 1 - 1.10T + 29T^{2} \) |
| 31 | \( 1 + 4.19T + 31T^{2} \) |
| 37 | \( 1 + 0.245T + 37T^{2} \) |
| 41 | \( 1 + 7.23T + 41T^{2} \) |
| 43 | \( 1 + 7.67T + 43T^{2} \) |
| 47 | \( 1 + 3.82T + 47T^{2} \) |
| 53 | \( 1 + 7.47T + 53T^{2} \) |
| 59 | \( 1 - 6.85T + 59T^{2} \) |
| 61 | \( 1 + 3.37T + 61T^{2} \) |
| 67 | \( 1 + 9.57T + 67T^{2} \) |
| 71 | \( 1 + 0.576T + 71T^{2} \) |
| 73 | \( 1 - 5.33T + 73T^{2} \) |
| 79 | \( 1 - 6.23T + 79T^{2} \) |
| 83 | \( 1 + 4.07T + 83T^{2} \) |
| 89 | \( 1 - 6.46T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−9.924549471617796332262500994941, −9.583585427432778695523747435070, −8.253760316686257791040291356920, −7.73361933927410955078460037748, −6.90592032690430360733658389615, −5.46698421603451104779260461323, −4.68140921102196253909645667842, −3.50169197808424830648642498962, −1.53482211517777572812253890228, 0,
1.53482211517777572812253890228, 3.50169197808424830648642498962, 4.68140921102196253909645667842, 5.46698421603451104779260461323, 6.90592032690430360733658389615, 7.73361933927410955078460037748, 8.253760316686257791040291356920, 9.583585427432778695523747435070, 9.924549471617796332262500994941
