L(s) = 1 | + 2-s + 4-s − 2.43·7-s + 8-s + 4.08·11-s + 3.79·13-s − 2.43·14-s + 16-s − 3.73·17-s + 19-s + 4.08·22-s − 0.351·23-s + 3.79·26-s − 2.43·28-s + 6.73·29-s + 9.34·31-s + 32-s − 3.73·34-s − 6.17·37-s + 38-s + 4.05·41-s + 0.913·43-s + 4.08·44-s − 0.351·46-s + 5.52·47-s − 1.05·49-s + 3.79·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.921·7-s + 0.353·8-s + 1.23·11-s + 1.05·13-s − 0.651·14-s + 0.250·16-s − 0.905·17-s + 0.229·19-s + 0.871·22-s − 0.0733·23-s + 0.743·26-s − 0.460·28-s + 1.25·29-s + 1.67·31-s + 0.176·32-s − 0.640·34-s − 1.01·37-s + 0.162·38-s + 0.633·41-s + 0.139·43-s + 0.616·44-s − 0.0518·46-s + 0.805·47-s − 0.150·49-s + 0.525·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.447048211\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.447048211\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 - 4.08T + 11T^{2} \) |
| 13 | \( 1 - 3.79T + 13T^{2} \) |
| 17 | \( 1 + 3.73T + 17T^{2} \) |
| 23 | \( 1 + 0.351T + 23T^{2} \) |
| 29 | \( 1 - 6.73T + 29T^{2} \) |
| 31 | \( 1 - 9.34T + 31T^{2} \) |
| 37 | \( 1 + 6.17T + 37T^{2} \) |
| 41 | \( 1 - 4.05T + 41T^{2} \) |
| 43 | \( 1 - 0.913T + 43T^{2} \) |
| 47 | \( 1 - 5.52T + 47T^{2} \) |
| 53 | \( 1 + 6.96T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 7.61T + 67T^{2} \) |
| 71 | \( 1 - 9.73T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 3.17T + 79T^{2} \) |
| 83 | \( 1 + 2.90T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−7.66314433719462315108476863437, −6.71356411083531853867863444944, −6.35725201065930785945557507554, −6.00221017337040224056485351019, −4.77602086070496884053760557578, −4.30213120317229113320878845579, −3.44493295092591757962225673317, −2.95571960065060410089340804715, −1.82383135188140536533507806397, −0.837686227781796766820280731618,
0.837686227781796766820280731618, 1.82383135188140536533507806397, 2.95571960065060410089340804715, 3.44493295092591757962225673317, 4.30213120317229113320878845579, 4.77602086070496884053760557578, 6.00221017337040224056485351019, 6.35725201065930785945557507554, 6.71356411083531853867863444944, 7.66314433719462315108476863437