| L(s) = 1 | − 2.62·2-s − 3.16·3-s + 4.88·4-s + 5-s + 8.29·6-s − 7.56·8-s + 6.99·9-s − 2.62·10-s + 11-s − 15.4·12-s + 2.92·13-s − 3.16·15-s + 10.0·16-s − 4.34·17-s − 18.3·18-s + 4.16·19-s + 4.88·20-s − 2.62·22-s + 7.46·23-s + 23.9·24-s + 25-s − 7.67·26-s − 12.6·27-s − 1.51·29-s + 8.29·30-s − 7.31·31-s − 11.3·32-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 1.82·3-s + 2.44·4-s + 0.447·5-s + 3.38·6-s − 2.67·8-s + 2.33·9-s − 0.829·10-s + 0.301·11-s − 4.45·12-s + 0.810·13-s − 0.816·15-s + 2.52·16-s − 1.05·17-s − 4.32·18-s + 0.955·19-s + 1.09·20-s − 0.559·22-s + 1.55·23-s + 4.88·24-s + 0.200·25-s − 1.50·26-s − 2.42·27-s − 0.280·29-s + 1.51·30-s − 1.31·31-s − 2.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 + 3.16T + 3T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 + 4.34T + 17T^{2} \) |
| 19 | \( 1 - 4.16T + 19T^{2} \) |
| 23 | \( 1 - 7.46T + 23T^{2} \) |
| 29 | \( 1 + 1.51T + 29T^{2} \) |
| 31 | \( 1 + 7.31T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 9.30T + 41T^{2} \) |
| 43 | \( 1 - 6.10T + 43T^{2} \) |
| 47 | \( 1 + 0.486T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 6.41T + 61T^{2} \) |
| 67 | \( 1 - 5.11T + 67T^{2} \) |
| 71 | \( 1 - 4.63T + 71T^{2} \) |
| 73 | \( 1 - 5.56T + 73T^{2} \) |
| 79 | \( 1 - 1.62T + 79T^{2} \) |
| 83 | \( 1 + 3.35T + 83T^{2} \) |
| 89 | \( 1 + 2.74T + 89T^{2} \) |
| 97 | \( 1 - 4.34T + 97T^{2} \) |
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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
−8.746051058478489590095821918937, −7.57801537420053173074875978427, −6.84066469258657194597991227619, −6.53804059245411547021269979362, −5.63475726178013020009447002212, −4.92855477850642712061552086799, −3.42139623771665067661089331097, −1.83055435499033680280819180411, −1.13963713169875181307397835449, 0,
1.13963713169875181307397835449, 1.83055435499033680280819180411, 3.42139623771665067661089331097, 4.92855477850642712061552086799, 5.63475726178013020009447002212, 6.53804059245411547021269979362, 6.84066469258657194597991227619, 7.57801537420053173074875978427, 8.746051058478489590095821918937
