L(s) = 1 | + 2.34·2-s − 3-s + 3.48·4-s + 5-s − 2.34·6-s − 7-s + 3.48·8-s + 9-s + 2.34·10-s − 11-s − 3.48·12-s + 3.14·13-s − 2.34·14-s − 15-s + 1.19·16-s + 6.48·17-s + 2.34·18-s + 7.17·19-s + 3.48·20-s + 21-s − 2.34·22-s + 2.19·23-s − 3.48·24-s + 25-s + 7.37·26-s − 27-s − 3.48·28-s + ⋯ |
L(s) = 1 | + 1.65·2-s − 0.577·3-s + 1.74·4-s + 0.447·5-s − 0.956·6-s − 0.377·7-s + 1.23·8-s + 0.333·9-s + 0.740·10-s − 0.301·11-s − 1.00·12-s + 0.872·13-s − 0.626·14-s − 0.258·15-s + 0.299·16-s + 1.57·17-s + 0.552·18-s + 1.64·19-s + 0.780·20-s + 0.218·21-s − 0.499·22-s + 0.458·23-s − 0.712·24-s + 0.200·25-s + 1.44·26-s − 0.192·27-s − 0.659·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.874598782\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.874598782\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 2.34T + 2T^{2} \) |
| 13 | \( 1 - 3.14T + 13T^{2} \) |
| 17 | \( 1 - 6.48T + 17T^{2} \) |
| 19 | \( 1 - 7.17T + 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 - 0.949T + 29T^{2} \) |
| 31 | \( 1 + 6.12T + 31T^{2} \) |
| 37 | \( 1 - 0.853T + 37T^{2} \) |
| 41 | \( 1 - 3.53T + 41T^{2} \) |
| 43 | \( 1 + 5.63T + 43T^{2} \) |
| 47 | \( 1 + 1.53T + 47T^{2} \) |
| 53 | \( 1 + 6.15T + 53T^{2} \) |
| 59 | \( 1 + 5.63T + 59T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 8.22T + 71T^{2} \) |
| 73 | \( 1 + 4.68T + 73T^{2} \) |
| 79 | \( 1 - 2.51T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 + 5.00T + 97T^{2} \) |
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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
−10.01389216270181748379046785519, −9.169616839016537032560116942572, −7.74771173946090028548696667742, −6.96517124297829632728909276927, −5.97548076295552579664459818081, −5.57141400112530864868395516464, −4.81458666021509121619292655991, −3.58632569765157522957911669834, −2.99836757066342688132379713773, −1.39686093246280100302829021436,
1.39686093246280100302829021436, 2.99836757066342688132379713773, 3.58632569765157522957911669834, 4.81458666021509121619292655991, 5.57141400112530864868395516464, 5.97548076295552579664459818081, 6.96517124297829632728909276927, 7.74771173946090028548696667742, 9.169616839016537032560116942572, 10.01389216270181748379046785519