| L(s) = 1 | − 3-s − 5-s − 1.23·7-s + 9-s − 3.23·13-s + 15-s + 4.47·17-s + 6.47·19-s + 1.23·21-s − 4·23-s + 25-s − 27-s − 7.23·29-s − 31-s + 1.23·35-s + 4.76·37-s + 3.23·39-s − 6·41-s + 6.47·43-s − 45-s + 8.94·47-s − 5.47·49-s − 4.47·51-s + 8.47·53-s − 6.47·57-s + 6.76·59-s − 12.4·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.467·7-s + 0.333·9-s − 0.897·13-s + 0.258·15-s + 1.08·17-s + 1.48·19-s + 0.269·21-s − 0.834·23-s + 0.200·25-s − 0.192·27-s − 1.34·29-s − 0.179·31-s + 0.208·35-s + 0.783·37-s + 0.518·39-s − 0.937·41-s + 0.986·43-s − 0.149·45-s + 1.30·47-s − 0.781·49-s − 0.626·51-s + 1.16·53-s − 0.857·57-s + 0.880·59-s − 1.59·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 7.23T + 29T^{2} \) |
| 37 | \( 1 - 4.76T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 6.76T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 + 9.23T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 4.94T + 83T^{2} \) |
| 89 | \( 1 + 7.23T + 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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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
−7.82351406466869634282047463533, −7.50933929473104756363391763666, −6.75756435225147561006332244579, −5.67746497518047663249582866714, −5.37940507506516526812860901885, −4.27157908298487974828005476628, −3.53045995532553252199680384501, −2.59024059186908339093324335047, −1.23441897082536577684323572463, 0,
1.23441897082536577684323572463, 2.59024059186908339093324335047, 3.53045995532553252199680384501, 4.27157908298487974828005476628, 5.37940507506516526812860901885, 5.67746497518047663249582866714, 6.75756435225147561006332244579, 7.50933929473104756363391763666, 7.82351406466869634282047463533
