| L(s) = 1 | − 1.34·2-s − 3-s − 0.184·4-s + 2.53·5-s + 1.34·6-s − 0.184·7-s + 2.94·8-s + 9-s − 3.41·10-s − 3.57·11-s + 0.184·12-s − 6.53·13-s + 0.248·14-s − 2.53·15-s − 3.59·16-s − 1.34·18-s + 4.63·19-s − 0.467·20-s + 0.184·21-s + 4.81·22-s + 3.10·23-s − 2.94·24-s + 1.41·25-s + 8.80·26-s − 27-s + 0.0341·28-s − 6.35·29-s + ⋯ |
| L(s) = 1 | − 0.952·2-s − 0.577·3-s − 0.0923·4-s + 1.13·5-s + 0.550·6-s − 0.0698·7-s + 1.04·8-s + 0.333·9-s − 1.07·10-s − 1.07·11-s + 0.0533·12-s − 1.81·13-s + 0.0665·14-s − 0.653·15-s − 0.899·16-s − 0.317·18-s + 1.06·19-s − 0.104·20-s + 0.0403·21-s + 1.02·22-s + 0.647·23-s − 0.600·24-s + 0.282·25-s + 1.72·26-s − 0.192·27-s + 0.00645·28-s − 1.18·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.34T + 2T^{2} \) |
| 5 | \( 1 - 2.53T + 5T^{2} \) |
| 7 | \( 1 + 0.184T + 7T^{2} \) |
| 11 | \( 1 + 3.57T + 11T^{2} \) |
| 13 | \( 1 + 6.53T + 13T^{2} \) |
| 19 | \( 1 - 4.63T + 19T^{2} \) |
| 23 | \( 1 - 3.10T + 23T^{2} \) |
| 29 | \( 1 + 6.35T + 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 - 8.39T + 37T^{2} \) |
| 41 | \( 1 + 7.18T + 41T^{2} \) |
| 43 | \( 1 + 4.93T + 43T^{2} \) |
| 47 | \( 1 + 9.04T + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 + 6.55T + 59T^{2} \) |
| 61 | \( 1 + 2.98T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 3.61T + 71T^{2} \) |
| 73 | \( 1 + 8.41T + 73T^{2} \) |
| 79 | \( 1 - 7.82T + 79T^{2} \) |
| 83 | \( 1 - 2.51T + 83T^{2} \) |
| 89 | \( 1 - 1.32T + 89T^{2} \) |
| 97 | \( 1 - 8.75T + 97T^{2} \) |
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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
−9.820810055570041602011283593184, −9.268938293697234469279451117048, −7.953397214309557223302374535810, −7.41980140553918696218228691757, −6.32932987634230956949371925339, −5.14463402777251102131478554317, −4.83582629529553416203931504949, −2.83773364073081434575576127361, −1.62206204155627742971173896306, 0,
1.62206204155627742971173896306, 2.83773364073081434575576127361, 4.83582629529553416203931504949, 5.14463402777251102131478554317, 6.32932987634230956949371925339, 7.41980140553918696218228691757, 7.953397214309557223302374535810, 9.268938293697234469279451117048, 9.820810055570041602011283593184
