| L(s) = 1 | − 2-s + 3.30·3-s + 4-s − 2.30·5-s − 3.30·6-s − 2.60·7-s − 8-s + 7.90·9-s + 2.30·10-s − 2.30·11-s + 3.30·12-s + 1.30·13-s + 2.60·14-s − 7.60·15-s + 16-s − 6·17-s − 7.90·18-s + 2·19-s − 2.30·20-s − 8.60·21-s + 2.30·22-s + 3.90·23-s − 3.30·24-s + 0.302·25-s − 1.30·26-s + 16.2·27-s − 2.60·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.90·3-s + 0.5·4-s − 1.02·5-s − 1.34·6-s − 0.984·7-s − 0.353·8-s + 2.63·9-s + 0.728·10-s − 0.694·11-s + 0.953·12-s + 0.361·13-s + 0.696·14-s − 1.96·15-s + 0.250·16-s − 1.45·17-s − 1.86·18-s + 0.458·19-s − 0.514·20-s − 1.87·21-s + 0.490·22-s + 0.814·23-s − 0.674·24-s + 0.0605·25-s − 0.255·26-s + 3.11·27-s − 0.492·28-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
| \(L(1)\) |
\(\approx\) |
\(0.9642948923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9642948923\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.30T + 3T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 7 | \( 1 + 2.60T + 7T^{2} \) |
| 11 | \( 1 + 2.30T + 11T^{2} \) |
| 13 | \( 1 - 1.30T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 3.90T + 23T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 + 0.302T + 31T^{2} \) |
| 41 | \( 1 - 9.90T + 41T^{2} \) |
| 43 | \( 1 - 0.605T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 + 3.51T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 9.11T + 79T^{2} \) |
| 83 | \( 1 - 2.78T + 83T^{2} \) |
| 89 | \( 1 + 9.21T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−14.85715955519318499835389416333, −13.45578658411617519237182903232, −12.72332663292485060593600802018, −11.00558376992052385392883780627, −9.654858831283765204535950149774, −8.841253477501806828214993356905, −7.86588361361950631323811500192, −6.96682723610503755952759087238, −3.93060845224348257138198386426, −2.69741759734566796341285138193,
2.69741759734566796341285138193, 3.93060845224348257138198386426, 6.96682723610503755952759087238, 7.86588361361950631323811500192, 8.841253477501806828214993356905, 9.654858831283765204535950149774, 11.00558376992052385392883780627, 12.72332663292485060593600802018, 13.45578658411617519237182903232, 14.85715955519318499835389416333
