| L(s) = 1 | − 1.50·2-s − 0.619·3-s + 0.277·4-s + 5-s + 0.934·6-s + 1.62·7-s + 2.59·8-s − 2.61·9-s − 1.50·10-s − 11-s − 0.171·12-s − 6.84·13-s − 2.45·14-s − 0.619·15-s − 4.47·16-s − 4.30·17-s + 3.94·18-s − 4.24·19-s + 0.277·20-s − 1.00·21-s + 1.50·22-s − 3.05·23-s − 1.61·24-s + 25-s + 10.3·26-s + 3.47·27-s + 0.451·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.357·3-s + 0.138·4-s + 0.447·5-s + 0.381·6-s + 0.615·7-s + 0.919·8-s − 0.872·9-s − 0.477·10-s − 0.301·11-s − 0.0496·12-s − 1.89·13-s − 0.656·14-s − 0.159·15-s − 1.11·16-s − 1.04·17-s + 0.930·18-s − 0.973·19-s + 0.0620·20-s − 0.219·21-s + 0.321·22-s − 0.636·23-s − 0.328·24-s + 0.200·25-s + 2.02·26-s + 0.669·27-s + 0.0853·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3466912972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3466912972\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 1.50T + 2T^{2} \) |
| 3 | \( 1 + 0.619T + 3T^{2} \) |
| 7 | \( 1 - 1.62T + 7T^{2} \) |
| 13 | \( 1 + 6.84T + 13T^{2} \) |
| 17 | \( 1 + 4.30T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 + 3.05T + 23T^{2} \) |
| 29 | \( 1 + 8.79T + 29T^{2} \) |
| 31 | \( 1 - 3.50T + 31T^{2} \) |
| 37 | \( 1 + 8.29T + 37T^{2} \) |
| 41 | \( 1 + 5.21T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 - 8.33T + 47T^{2} \) |
| 53 | \( 1 + 2.00T + 53T^{2} \) |
| 59 | \( 1 - 3.57T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 7.09T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 79 | \( 1 + 3.17T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 6.31T + 97T^{2} \) |
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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
−8.525699152367010104233755002485, −7.87372318299411868473797278722, −7.16659318898464605647422768428, −6.41163474070552564891971090229, −5.29032224756125017690579422126, −4.95065843406481033955359810601, −3.97769461272928579288471824176, −2.37910849738953026424745301289, −2.00582195670997153118899988522, −0.38212475000299499906748588297,
0.38212475000299499906748588297, 2.00582195670997153118899988522, 2.37910849738953026424745301289, 3.97769461272928579288471824176, 4.95065843406481033955359810601, 5.29032224756125017690579422126, 6.41163474070552564891971090229, 7.16659318898464605647422768428, 7.87372318299411868473797278722, 8.525699152367010104233755002485
