L(s) = 1 | − 5-s + 0.911·7-s − 2.91·11-s − 6.25·13-s − 4·17-s + 19-s − 5.34·23-s + 25-s + 0.911·29-s − 2·31-s − 0.911·35-s + 4.43·37-s − 6.43·41-s + 8.25·43-s + 5.34·47-s − 6.16·49-s − 3.34·53-s + 2.91·55-s − 5.52·59-s + 13.1·61-s + 6.25·65-s + 4·67-s + 16.5·71-s + 6·73-s − 2.65·77-s − 10.9·79-s + 3.82·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.344·7-s − 0.877·11-s − 1.73·13-s − 0.970·17-s + 0.229·19-s − 1.11·23-s + 0.200·25-s + 0.169·29-s − 0.359·31-s − 0.154·35-s + 0.729·37-s − 1.00·41-s + 1.25·43-s + 0.779·47-s − 0.881·49-s − 0.459·53-s + 0.392·55-s − 0.719·59-s + 1.68·61-s + 0.776·65-s + 0.488·67-s + 1.96·71-s + 0.702·73-s − 0.302·77-s − 1.23·79-s + 0.419·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9838482588\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9838482588\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 0.911T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 + 6.25T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 23 | \( 1 + 5.34T + 23T^{2} \) |
| 29 | \( 1 - 0.911T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 4.43T + 37T^{2} \) |
| 41 | \( 1 + 6.43T + 41T^{2} \) |
| 43 | \( 1 - 8.25T + 43T^{2} \) |
| 47 | \( 1 - 5.34T + 47T^{2} \) |
| 53 | \( 1 + 3.34T + 53T^{2} \) |
| 59 | \( 1 + 5.52T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 16.5T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 3.82T + 83T^{2} \) |
| 89 | \( 1 - 2.43T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−7.899893048434733595849118362531, −7.38643402877379335984602321881, −6.70948586618996500422522391632, −5.77432144885595697517727758774, −4.97525233285085837104077820219, −4.53945119716778137561510076577, −3.62185793346839368812218234722, −2.55463245235526571176693829221, −2.06383609804152082296926841603, −0.47729104476777837586869306108,
0.47729104476777837586869306108, 2.06383609804152082296926841603, 2.55463245235526571176693829221, 3.62185793346839368812218234722, 4.53945119716778137561510076577, 4.97525233285085837104077820219, 5.77432144885595697517727758774, 6.70948586618996500422522391632, 7.38643402877379335984602321881, 7.899893048434733595849118362531