| L(s) = 1 | − 1.31·2-s + 2.15·3-s − 0.275·4-s − 2.87·5-s − 2.82·6-s + 2.98·8-s + 1.64·9-s + 3.76·10-s + 1.59·11-s − 0.592·12-s − 4.65·13-s − 6.18·15-s − 3.37·16-s − 1.70·17-s − 2.15·18-s − 4.60·19-s + 0.789·20-s − 2.08·22-s − 0.103·23-s + 6.43·24-s + 3.23·25-s + 6.11·26-s − 2.92·27-s − 7.33·29-s + 8.12·30-s + 5.98·31-s − 1.54·32-s + ⋯ |
| L(s) = 1 | − 0.928·2-s + 1.24·3-s − 0.137·4-s − 1.28·5-s − 1.15·6-s + 1.05·8-s + 0.547·9-s + 1.19·10-s + 0.479·11-s − 0.171·12-s − 1.29·13-s − 1.59·15-s − 0.843·16-s − 0.413·17-s − 0.508·18-s − 1.05·19-s + 0.176·20-s − 0.445·22-s − 0.0216·23-s + 1.31·24-s + 0.647·25-s + 1.19·26-s − 0.563·27-s − 1.36·29-s + 1.48·30-s + 1.07·31-s − 0.273·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7191747951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7191747951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 1.31T + 2T^{2} \) |
| 3 | \( 1 - 2.15T + 3T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 11 | \( 1 - 1.59T + 11T^{2} \) |
| 13 | \( 1 + 4.65T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 + 4.60T + 19T^{2} \) |
| 23 | \( 1 + 0.103T + 23T^{2} \) |
| 29 | \( 1 + 7.33T + 29T^{2} \) |
| 31 | \( 1 - 5.98T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 2.98T + 43T^{2} \) |
| 47 | \( 1 + 5.25T + 47T^{2} \) |
| 53 | \( 1 + 8.35T + 53T^{2} \) |
| 59 | \( 1 - 6.58T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 7.44T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 9.12T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 - 7.66T + 83T^{2} \) |
| 89 | \( 1 - 8.35T + 89T^{2} \) |
| 97 | \( 1 - 7.47T + 97T^{2} \) |
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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
−8.110051537416789059660263622790, −7.57893139198185424366857899565, −7.24245952691990628087879076920, −6.15409900092438024065781558474, −4.80089929938206105990866932540, −4.26512057402031822241227683083, −3.66509094645901524622260237478, −2.65080329051502542681441149480, −1.86852088572302379016752344057, −0.46567960174872686950206619942,
0.46567960174872686950206619942, 1.86852088572302379016752344057, 2.65080329051502542681441149480, 3.66509094645901524622260237478, 4.26512057402031822241227683083, 4.80089929938206105990866932540, 6.15409900092438024065781558474, 7.24245952691990628087879076920, 7.57893139198185424366857899565, 8.110051537416789059660263622790
