L(s) = 1 | + 2-s + 4-s − 5-s + 0.712·7-s + 8-s − 10-s − 1.86·11-s − 5.74·13-s + 0.712·14-s + 16-s − 4.32·17-s + 4.76·19-s − 20-s − 1.86·22-s + 6.50·23-s + 25-s − 5.74·26-s + 0.712·28-s − 1.95·29-s + 5.88·31-s + 32-s − 4.32·34-s − 0.712·35-s + 7.35·37-s + 4.76·38-s − 40-s + 0.135·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.269·7-s + 0.353·8-s − 0.316·10-s − 0.562·11-s − 1.59·13-s + 0.190·14-s + 0.250·16-s − 1.04·17-s + 1.09·19-s − 0.223·20-s − 0.397·22-s + 1.35·23-s + 0.200·25-s − 1.12·26-s + 0.134·28-s − 0.362·29-s + 1.05·31-s + 0.176·32-s − 0.741·34-s − 0.120·35-s + 1.20·37-s + 0.772·38-s − 0.158·40-s + 0.0211·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + T \) |
good | 7 | \( 1 - 0.712T + 7T^{2} \) |
| 11 | \( 1 + 1.86T + 11T^{2} \) |
| 13 | \( 1 + 5.74T + 13T^{2} \) |
| 17 | \( 1 + 4.32T + 17T^{2} \) |
| 19 | \( 1 - 4.76T + 19T^{2} \) |
| 23 | \( 1 - 6.50T + 23T^{2} \) |
| 29 | \( 1 + 1.95T + 29T^{2} \) |
| 31 | \( 1 - 5.88T + 31T^{2} \) |
| 37 | \( 1 - 7.35T + 37T^{2} \) |
| 41 | \( 1 - 0.135T + 41T^{2} \) |
| 43 | \( 1 + 6.18T + 43T^{2} \) |
| 47 | \( 1 + 3.28T + 47T^{2} \) |
| 53 | \( 1 + 8.55T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 71 | \( 1 + 1.55T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 + 0.0873T + 79T^{2} \) |
| 83 | \( 1 - 5.08T + 83T^{2} \) |
| 89 | \( 1 + 2.27T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−7.75792102753490215236252355827, −6.94905088908442574823113193983, −6.39078264288108531608047630686, −5.22628665886938795346323303512, −4.88896781510742588311146271605, −4.26654438182385578270995232431, −3.03261479011271626939439228239, −2.69484860594087387916026380060, −1.47111625580531283811875517877, 0,
1.47111625580531283811875517877, 2.69484860594087387916026380060, 3.03261479011271626939439228239, 4.26654438182385578270995232431, 4.88896781510742588311146271605, 5.22628665886938795346323303512, 6.39078264288108531608047630686, 6.94905088908442574823113193983, 7.75792102753490215236252355827