| L(s) = 1 | + 2.78·2-s − 3-s + 5.73·4-s − 0.825·5-s − 2.78·6-s + 10.3·8-s + 9-s − 2.29·10-s − 11-s − 5.73·12-s + 0.296·13-s + 0.825·15-s + 17.4·16-s + 6.69·17-s + 2.78·18-s + 2.83·19-s − 4.73·20-s − 2.78·22-s − 3.96·23-s − 10.3·24-s − 4.31·25-s + 0.825·26-s − 27-s − 0.484·29-s + 2.29·30-s + 7.33·31-s + 27.7·32-s + ⋯ |
| L(s) = 1 | + 1.96·2-s − 0.577·3-s + 2.86·4-s − 0.369·5-s − 1.13·6-s + 3.67·8-s + 0.333·9-s − 0.726·10-s − 0.301·11-s − 1.65·12-s + 0.0823·13-s + 0.213·15-s + 4.36·16-s + 1.62·17-s + 0.655·18-s + 0.650·19-s − 1.05·20-s − 0.593·22-s − 0.827·23-s − 2.12·24-s − 0.863·25-s + 0.161·26-s − 0.192·27-s − 0.0900·29-s + 0.419·30-s + 1.31·31-s + 4.90·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.113938714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.113938714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.78T + 2T^{2} \) |
| 5 | \( 1 + 0.825T + 5T^{2} \) |
| 13 | \( 1 - 0.296T + 13T^{2} \) |
| 17 | \( 1 - 6.69T + 17T^{2} \) |
| 19 | \( 1 - 2.83T + 19T^{2} \) |
| 23 | \( 1 + 3.96T + 23T^{2} \) |
| 29 | \( 1 + 0.484T + 29T^{2} \) |
| 31 | \( 1 - 7.33T + 31T^{2} \) |
| 37 | \( 1 + 5.73T + 37T^{2} \) |
| 41 | \( 1 + 0.645T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 + 7.73T + 47T^{2} \) |
| 53 | \( 1 - 7.11T + 53T^{2} \) |
| 59 | \( 1 - 1.15T + 59T^{2} \) |
| 61 | \( 1 + 5.26T + 61T^{2} \) |
| 67 | \( 1 - 3.01T + 67T^{2} \) |
| 71 | \( 1 - 3.58T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 + 4.32T + 79T^{2} \) |
| 83 | \( 1 + 2.37T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−9.899661583837573387380894570744, −8.068509465854938727912363528603, −7.52845750586556426708618881432, −6.67197145223185987583648854003, −5.77104301448189051307873321554, −5.37449659216949371249161301902, −4.39445461628573660681979484034, −3.65144974776885130917510360130, −2.77368242328681465194787690714, −1.42924797399794833768650304110,
1.42924797399794833768650304110, 2.77368242328681465194787690714, 3.65144974776885130917510360130, 4.39445461628573660681979484034, 5.37449659216949371249161301902, 5.77104301448189051307873321554, 6.67197145223185987583648854003, 7.52845750586556426708618881432, 8.068509465854938727912363528603, 9.899661583837573387380894570744
