L(s) = 1 | − 0.347·2-s − 3-s − 1.87·4-s + 5-s + 0.347·6-s − 4.57·7-s + 1.34·8-s + 9-s − 0.347·10-s + 5.17·11-s + 1.87·12-s − 4.70·13-s + 1.59·14-s − 15-s + 3.28·16-s + 3.74·17-s − 0.347·18-s − 1.63·19-s − 1.87·20-s + 4.57·21-s − 1.79·22-s − 4.05·23-s − 1.34·24-s + 25-s + 1.63·26-s − 27-s + 8.60·28-s + ⋯ |
L(s) = 1 | − 0.245·2-s − 0.577·3-s − 0.939·4-s + 0.447·5-s + 0.141·6-s − 1.73·7-s + 0.476·8-s + 0.333·9-s − 0.109·10-s + 1.55·11-s + 0.542·12-s − 1.30·13-s + 0.425·14-s − 0.258·15-s + 0.822·16-s + 0.908·17-s − 0.0819·18-s − 0.374·19-s − 0.420·20-s + 0.999·21-s − 0.383·22-s − 0.844·23-s − 0.275·24-s + 0.200·25-s + 0.320·26-s − 0.192·27-s + 1.62·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6011408388\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6011408388\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 + 0.347T + 2T^{2} \) |
| 7 | \( 1 + 4.57T + 7T^{2} \) |
| 11 | \( 1 - 5.17T + 11T^{2} \) |
| 13 | \( 1 + 4.70T + 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 + 1.63T + 19T^{2} \) |
| 23 | \( 1 + 4.05T + 23T^{2} \) |
| 29 | \( 1 + 4.36T + 29T^{2} \) |
| 31 | \( 1 - 0.267T + 31T^{2} \) |
| 37 | \( 1 + 2.19T + 37T^{2} \) |
| 41 | \( 1 - 2.00T + 41T^{2} \) |
| 43 | \( 1 - 3.41T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 7.89T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 2.07T + 61T^{2} \) |
| 67 | \( 1 - 7.80T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 3.67T + 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.158511133151189704542914508789, −7.11400214828245750425463876237, −6.74995595168552703450766935263, −5.82974491621603311232218100174, −5.45466347515848706828384759033, −4.28678016708427896211620189281, −3.83370764842131983095271089438, −2.89177291067313513589937086699, −1.59448685542242147797035589710, −0.44794652462043877397975332926,
0.44794652462043877397975332926, 1.59448685542242147797035589710, 2.89177291067313513589937086699, 3.83370764842131983095271089438, 4.28678016708427896211620189281, 5.45466347515848706828384759033, 5.82974491621603311232218100174, 6.74995595168552703450766935263, 7.11400214828245750425463876237, 8.158511133151189704542914508789