| L(s) = 1 | − 2.08·2-s − 2.29·3-s + 2.34·4-s − 0.933·5-s + 4.77·6-s − 0.725·8-s + 2.25·9-s + 1.94·10-s + 2.78·11-s − 5.38·12-s + 0.193·13-s + 2.14·15-s − 3.18·16-s + 2.51·17-s − 4.70·18-s + 7.64·19-s − 2.19·20-s − 5.81·22-s − 5.15·23-s + 1.66·24-s − 4.12·25-s − 0.404·26-s + 1.70·27-s + 5.69·29-s − 4.46·30-s − 0.00490·31-s + 8.08·32-s + ⋯ |
| L(s) = 1 | − 1.47·2-s − 1.32·3-s + 1.17·4-s − 0.417·5-s + 1.95·6-s − 0.256·8-s + 0.751·9-s + 0.615·10-s + 0.840·11-s − 1.55·12-s + 0.0537·13-s + 0.552·15-s − 0.795·16-s + 0.610·17-s − 1.10·18-s + 1.75·19-s − 0.490·20-s − 1.23·22-s − 1.07·23-s + 0.339·24-s − 0.825·25-s − 0.0792·26-s + 0.328·27-s + 1.05·29-s − 0.814·30-s − 0.000881·31-s + 1.42·32-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
| \(L(1)\) |
\(\approx\) |
\(0.5788725118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5788725118\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.08T + 2T^{2} \) |
| 3 | \( 1 + 2.29T + 3T^{2} \) |
| 5 | \( 1 + 0.933T + 5T^{2} \) |
| 11 | \( 1 - 2.78T + 11T^{2} \) |
| 13 | \( 1 - 0.193T + 13T^{2} \) |
| 17 | \( 1 - 2.51T + 17T^{2} \) |
| 19 | \( 1 - 7.64T + 19T^{2} \) |
| 23 | \( 1 + 5.15T + 23T^{2} \) |
| 29 | \( 1 - 5.69T + 29T^{2} \) |
| 31 | \( 1 + 0.00490T + 31T^{2} \) |
| 37 | \( 1 - 6.30T + 37T^{2} \) |
| 41 | \( 1 + 0.548T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 5.48T + 59T^{2} \) |
| 61 | \( 1 - 9.56T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 + 9.73T + 79T^{2} \) |
| 83 | \( 1 - 0.0567T + 83T^{2} \) |
| 89 | \( 1 + 2.59T + 89T^{2} \) |
| 97 | \( 1 - 8.05T + 97T^{2} \) |
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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
−8.045895050052091022634203977974, −7.38491660704792884437613553276, −6.86631106603135920967454106247, −5.99642547133848751733355397317, −5.48171063847597253754724990949, −4.47223483002768736154924218852, −3.70896884079413655418919665108, −2.41058691876046576192240232619, −1.13876139066187313649647756390, −0.67109224792568096707800118798,
0.67109224792568096707800118798, 1.13876139066187313649647756390, 2.41058691876046576192240232619, 3.70896884079413655418919665108, 4.47223483002768736154924218852, 5.48171063847597253754724990949, 5.99642547133848751733355397317, 6.86631106603135920967454106247, 7.38491660704792884437613553276, 8.045895050052091022634203977974
