| L(s) = 1 | − 2-s + 1.44·3-s + 4-s − 1.26·5-s − 1.44·6-s − 7-s − 8-s − 0.901·9-s + 1.26·10-s − 3.22·11-s + 1.44·12-s − 2.31·13-s + 14-s − 1.83·15-s + 16-s + 3.60·17-s + 0.901·18-s + 5.33·19-s − 1.26·20-s − 1.44·21-s + 3.22·22-s − 7.66·23-s − 1.44·24-s − 3.40·25-s + 2.31·26-s − 5.65·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.836·3-s + 0.5·4-s − 0.565·5-s − 0.591·6-s − 0.377·7-s − 0.353·8-s − 0.300·9-s + 0.399·10-s − 0.972·11-s + 0.418·12-s − 0.642·13-s + 0.267·14-s − 0.472·15-s + 0.250·16-s + 0.873·17-s + 0.212·18-s + 1.22·19-s − 0.282·20-s − 0.316·21-s + 0.687·22-s − 1.59·23-s − 0.295·24-s − 0.680·25-s + 0.454·26-s − 1.08·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9707624933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9707624933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 5 | \( 1 + 1.26T + 5T^{2} \) |
| 11 | \( 1 + 3.22T + 11T^{2} \) |
| 13 | \( 1 + 2.31T + 13T^{2} \) |
| 17 | \( 1 - 3.60T + 17T^{2} \) |
| 19 | \( 1 - 5.33T + 19T^{2} \) |
| 23 | \( 1 + 7.66T + 23T^{2} \) |
| 29 | \( 1 + 0.214T + 29T^{2} \) |
| 31 | \( 1 + 5.47T + 31T^{2} \) |
| 37 | \( 1 - 0.154T + 37T^{2} \) |
| 41 | \( 1 - 3.69T + 41T^{2} \) |
| 43 | \( 1 - 7.50T + 43T^{2} \) |
| 47 | \( 1 - 0.101T + 47T^{2} \) |
| 53 | \( 1 - 9.39T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 5.95T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 1.96T + 83T^{2} \) |
| 89 | \( 1 + 0.350T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−8.016825818473592256024889582892, −7.56421415438349822257873881196, −7.15590239960064867187984817966, −5.71819653511185232793397119883, −5.58219117341381144345315401901, −4.15843072055506334389954040160, −3.43835579637760336759956630560, −2.71368245027600069458846135677, −2.00533838834854344082608309406, −0.52105020623183248076912944279,
0.52105020623183248076912944279, 2.00533838834854344082608309406, 2.71368245027600069458846135677, 3.43835579637760336759956630560, 4.15843072055506334389954040160, 5.58219117341381144345315401901, 5.71819653511185232793397119883, 7.15590239960064867187984817966, 7.56421415438349822257873881196, 8.016825818473592256024889582892
