L(s) = 1 | + 2-s + 0.618·3-s + 4-s − 2.85·5-s + 0.618·6-s + 1.23·7-s + 8-s − 2.61·9-s − 2.85·10-s − 3.61·11-s + 0.618·12-s + 3.85·13-s + 1.23·14-s − 1.76·15-s + 16-s + 4.47·17-s − 2.61·18-s − 4.47·19-s − 2.85·20-s + 0.763·21-s − 3.61·22-s − 3.85·23-s + 0.618·24-s + 3.14·25-s + 3.85·26-s − 3.47·27-s + 1.23·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.356·3-s + 0.5·4-s − 1.27·5-s + 0.252·6-s + 0.467·7-s + 0.353·8-s − 0.872·9-s − 0.902·10-s − 1.09·11-s + 0.178·12-s + 1.06·13-s + 0.330·14-s − 0.455·15-s + 0.250·16-s + 1.08·17-s − 0.617·18-s − 1.02·19-s − 0.638·20-s + 0.166·21-s − 0.771·22-s − 0.803·23-s + 0.126·24-s + 0.629·25-s + 0.755·26-s − 0.668·27-s + 0.233·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.242921967\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.242921967\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 + 3.61T + 11T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 + 3.85T + 23T^{2} \) |
| 29 | \( 1 - 6.32T + 29T^{2} \) |
| 31 | \( 1 - 9.61T + 31T^{2} \) |
| 41 | \( 1 - 7.38T + 41T^{2} \) |
| 43 | \( 1 + 0.763T + 43T^{2} \) |
| 47 | \( 1 - 3.23T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 9.23T + 59T^{2} \) |
| 61 | \( 1 - 8.38T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 4.09T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 5.52T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−14.55568504382690214010878001481, −13.64492861912870423155420813297, −12.35190252318154107531529859448, −11.48124010159742978757620231313, −10.49717959069935234629396118610, −8.350653004117880787826262769279, −7.86716069247191929120431403684, −6.01796754899817048898544080092, −4.44392201551849548294070730841, −3.04662819818334026192895002349,
3.04662819818334026192895002349, 4.44392201551849548294070730841, 6.01796754899817048898544080092, 7.86716069247191929120431403684, 8.350653004117880787826262769279, 10.49717959069935234629396118610, 11.48124010159742978757620231313, 12.35190252318154107531529859448, 13.64492861912870423155420813297, 14.55568504382690214010878001481