| L(s) = 1 | − 1.99·3-s + 2.83·5-s − 3.85·7-s + 0.991·9-s + 6.23·13-s − 5.65·15-s + 4.97·17-s − 19-s + 7.69·21-s − 6.56·23-s + 3.01·25-s + 4.01·27-s − 3.54·29-s + 6.81·31-s − 10.8·35-s − 5.04·37-s − 12.4·39-s − 7.13·41-s − 11.4·43-s + 2.80·45-s − 9.88·47-s + 7.82·49-s − 9.94·51-s + 3.38·53-s + 1.99·57-s − 6.79·59-s + 13.7·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.26·5-s − 1.45·7-s + 0.330·9-s + 1.72·13-s − 1.46·15-s + 1.20·17-s − 0.229·19-s + 1.67·21-s − 1.36·23-s + 0.602·25-s + 0.772·27-s − 0.657·29-s + 1.22·31-s − 1.84·35-s − 0.830·37-s − 1.99·39-s − 1.11·41-s − 1.75·43-s + 0.418·45-s − 1.44·47-s + 1.11·49-s − 1.39·51-s + 0.464·53-s + 0.264·57-s − 0.884·59-s + 1.75·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.99T + 3T^{2} \) |
| 5 | \( 1 - 2.83T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 23 | \( 1 + 6.56T + 23T^{2} \) |
| 29 | \( 1 + 3.54T + 29T^{2} \) |
| 31 | \( 1 - 6.81T + 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 + 7.13T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 9.88T + 47T^{2} \) |
| 53 | \( 1 - 3.38T + 53T^{2} \) |
| 59 | \( 1 + 6.79T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 9.47T + 67T^{2} \) |
| 71 | \( 1 + 8.56T + 71T^{2} \) |
| 73 | \( 1 + 2.78T + 73T^{2} \) |
| 79 | \( 1 + 0.163T + 79T^{2} \) |
| 83 | \( 1 - 7.82T + 83T^{2} \) |
| 89 | \( 1 + 3.81T + 89T^{2} \) |
| 97 | \( 1 - 9.00T + 97T^{2} \) |
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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
−6.94626495159125228114828501532, −6.39297656962125884359015277503, −6.04888693265127503151582447982, −5.63269964450716861081953392920, −4.87915277985790797932294048992, −3.63000616733655471488011898862, −3.25483146739568094249798004762, −2.00770855008546560525665609846, −1.13588015633752938544850691829, 0,
1.13588015633752938544850691829, 2.00770855008546560525665609846, 3.25483146739568094249798004762, 3.63000616733655471488011898862, 4.87915277985790797932294048992, 5.63269964450716861081953392920, 6.04888693265127503151582447982, 6.39297656962125884359015277503, 6.94626495159125228114828501532
