L(s) = 1 | − 2-s + 3-s + 4-s − 3.69·5-s − 6-s + 3.64·7-s − 8-s + 9-s + 3.69·10-s + 11-s + 12-s − 6.50·13-s − 3.64·14-s − 3.69·15-s + 16-s − 6.91·17-s − 18-s + 2.83·19-s − 3.69·20-s + 3.64·21-s − 22-s − 5.46·23-s − 24-s + 8.68·25-s + 6.50·26-s + 27-s + 3.64·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.65·5-s − 0.408·6-s + 1.37·7-s − 0.353·8-s + 0.333·9-s + 1.16·10-s + 0.301·11-s + 0.288·12-s − 1.80·13-s − 0.974·14-s − 0.955·15-s + 0.250·16-s − 1.67·17-s − 0.235·18-s + 0.650·19-s − 0.827·20-s + 0.795·21-s − 0.213·22-s − 1.13·23-s − 0.204·24-s + 1.73·25-s + 1.27·26-s + 0.192·27-s + 0.689·28-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
\(L(1)\) |
\(\approx\) |
\(1.059568366\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.059568366\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.69T + 5T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 13 | \( 1 + 6.50T + 13T^{2} \) |
| 17 | \( 1 + 6.91T + 17T^{2} \) |
| 19 | \( 1 - 2.83T + 19T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 + 2.39T + 29T^{2} \) |
| 31 | \( 1 - 3.76T + 31T^{2} \) |
| 37 | \( 1 - 1.16T + 37T^{2} \) |
| 41 | \( 1 - 4.51T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 7.72T + 47T^{2} \) |
| 53 | \( 1 + 5.76T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 67 | \( 1 - 0.969T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 1.00T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.283033367141813561557200905001, −7.74301570498097947708359511939, −7.43369580754750904951097216781, −6.67226574170522127061726132344, −5.27726282282051648349248900820, −4.38263031799625901022399225693, −4.06012085153734909625909884947, −2.70135392310973632177050323602, −2.03052996581321614699337247080, −0.62017909927224501615766842410,
0.62017909927224501615766842410, 2.03052996581321614699337247080, 2.70135392310973632177050323602, 4.06012085153734909625909884947, 4.38263031799625901022399225693, 5.27726282282051648349248900820, 6.67226574170522127061726132344, 7.43369580754750904951097216781, 7.74301570498097947708359511939, 8.283033367141813561557200905001