| L(s) = 1 | − 2-s + 3-s + 4-s − 1.62·5-s − 6-s − 4.09·7-s − 8-s + 9-s + 1.62·10-s − 11-s + 12-s + 2.05·13-s + 4.09·14-s − 1.62·15-s + 16-s − 6.07·17-s − 18-s + 8.57·19-s − 1.62·20-s − 4.09·21-s + 22-s − 8.01·23-s − 24-s − 2.34·25-s − 2.05·26-s + 27-s − 4.09·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.728·5-s − 0.408·6-s − 1.54·7-s − 0.353·8-s + 0.333·9-s + 0.515·10-s − 0.301·11-s + 0.288·12-s + 0.569·13-s + 1.09·14-s − 0.420·15-s + 0.250·16-s − 1.47·17-s − 0.235·18-s + 1.96·19-s − 0.364·20-s − 0.893·21-s + 0.213·22-s − 1.67·23-s − 0.204·24-s − 0.469·25-s − 0.402·26-s + 0.192·27-s − 0.773·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7906636501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7906636501\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 + 1.62T + 5T^{2} \) |
| 7 | \( 1 + 4.09T + 7T^{2} \) |
| 13 | \( 1 - 2.05T + 13T^{2} \) |
| 17 | \( 1 + 6.07T + 17T^{2} \) |
| 19 | \( 1 - 8.57T + 19T^{2} \) |
| 23 | \( 1 + 8.01T + 23T^{2} \) |
| 29 | \( 1 - 1.71T + 29T^{2} \) |
| 31 | \( 1 + 4.38T + 31T^{2} \) |
| 37 | \( 1 - 3.63T + 37T^{2} \) |
| 41 | \( 1 - 2.67T + 41T^{2} \) |
| 43 | \( 1 + 1.73T + 43T^{2} \) |
| 47 | \( 1 - 9.02T + 47T^{2} \) |
| 53 | \( 1 + 8.44T + 53T^{2} \) |
| 59 | \( 1 - 6.78T + 59T^{2} \) |
| 67 | \( 1 + 8.71T + 67T^{2} \) |
| 71 | \( 1 + 4.87T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 4.94T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.566128174546728644228659300948, −7.58679569304361691001922455155, −7.34676486904042190472210237317, −6.33022094834705844012647451997, −5.80657537115114780649784365064, −4.39745633283712581779086197603, −3.59479738287534794253280567729, −3.01103704302308926235174790304, −2.00642259728775169766992514351, −0.52437709068718499791329215528,
0.52437709068718499791329215528, 2.00642259728775169766992514351, 3.01103704302308926235174790304, 3.59479738287534794253280567729, 4.39745633283712581779086197603, 5.80657537115114780649784365064, 6.33022094834705844012647451997, 7.34676486904042190472210237317, 7.58679569304361691001922455155, 8.566128174546728644228659300948
