| L(s) = 1 | − 0.318·2-s + 3-s − 1.89·4-s − 2.74·5-s − 0.318·6-s + 7-s + 1.24·8-s + 9-s + 0.875·10-s + 3.63·11-s − 1.89·12-s − 2.24·13-s − 0.318·14-s − 2.74·15-s + 3.40·16-s + 2.79·17-s − 0.318·18-s − 7.10·19-s + 5.21·20-s + 21-s − 1.15·22-s + 3.52·23-s + 1.24·24-s + 2.55·25-s + 0.714·26-s + 27-s − 1.89·28-s + ⋯ |
| L(s) = 1 | − 0.225·2-s + 0.577·3-s − 0.949·4-s − 1.22·5-s − 0.130·6-s + 0.377·7-s + 0.438·8-s + 0.333·9-s + 0.276·10-s + 1.09·11-s − 0.548·12-s − 0.621·13-s − 0.0851·14-s − 0.709·15-s + 0.850·16-s + 0.678·17-s − 0.0750·18-s − 1.63·19-s + 1.16·20-s + 0.218·21-s − 0.246·22-s + 0.734·23-s + 0.253·24-s + 0.510·25-s + 0.140·26-s + 0.192·27-s − 0.358·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 0.318T + 2T^{2} \) |
| 5 | \( 1 + 2.74T + 5T^{2} \) |
| 11 | \( 1 - 3.63T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 - 2.79T + 17T^{2} \) |
| 19 | \( 1 + 7.10T + 19T^{2} \) |
| 23 | \( 1 - 3.52T + 23T^{2} \) |
| 29 | \( 1 + 6.25T + 29T^{2} \) |
| 31 | \( 1 + 1.62T + 31T^{2} \) |
| 37 | \( 1 + 1.79T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 7.45T + 43T^{2} \) |
| 47 | \( 1 + 0.600T + 47T^{2} \) |
| 53 | \( 1 - 1.87T + 53T^{2} \) |
| 59 | \( 1 - 2.04T + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 - 5.47T + 67T^{2} \) |
| 71 | \( 1 + 6.16T + 71T^{2} \) |
| 73 | \( 1 - 6.35T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 2.46T + 97T^{2} \) |
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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
−7.62424939319988692259450817805, −7.13164462850193643043275554096, −6.16799675116202920852015873098, −5.13584086032369809932316680576, −4.46230025951293003239978366762, −3.89403041445916348307208556012, −3.43234674516302826191974374125, −2.16961169757711591622488473785, −1.10831896919184100310562177645, 0,
1.10831896919184100310562177645, 2.16961169757711591622488473785, 3.43234674516302826191974374125, 3.89403041445916348307208556012, 4.46230025951293003239978366762, 5.13584086032369809932316680576, 6.16799675116202920852015873098, 7.13164462850193643043275554096, 7.62424939319988692259450817805
