| L(s) = 1 | + 1.29·2-s − 3·3-s − 6.32·4-s − 17.0·5-s − 3.88·6-s + 13.0·7-s − 18.5·8-s + 9·9-s − 22.0·10-s − 58.1·11-s + 18.9·12-s − 76.5·13-s + 16.9·14-s + 51.1·15-s + 26.5·16-s + 31.4·17-s + 11.6·18-s − 95.5·19-s + 107.·20-s − 39.2·21-s − 75.3·22-s − 84.0·23-s + 55.6·24-s + 165.·25-s − 99.1·26-s − 27·27-s − 82.6·28-s + ⋯ |
| L(s) = 1 | + 0.458·2-s − 0.577·3-s − 0.790·4-s − 1.52·5-s − 0.264·6-s + 0.706·7-s − 0.820·8-s + 0.333·9-s − 0.698·10-s − 1.59·11-s + 0.456·12-s − 1.63·13-s + 0.323·14-s + 0.880·15-s + 0.414·16-s + 0.448·17-s + 0.152·18-s − 1.15·19-s + 1.20·20-s − 0.407·21-s − 0.729·22-s − 0.762·23-s + 0.473·24-s + 1.32·25-s − 0.748·26-s − 0.192·27-s − 0.558·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3757658007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3757658007\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 157 | \( 1 + 157T \) |
| good | 2 | \( 1 - 1.29T + 8T^{2} \) |
| 5 | \( 1 + 17.0T + 125T^{2} \) |
| 7 | \( 1 - 13.0T + 343T^{2} \) |
| 11 | \( 1 + 58.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 76.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 95.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 84.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 242.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 59.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 350.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 428.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 416.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 448.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 288.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 119.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 514.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 978.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 285.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.06e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 374.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.36e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 136.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 972.T + 9.12e5T^{2} \) |
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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
−10.62854398650478205156915559147, −9.992054974021436185338001228082, −8.425450392327746416747695422336, −7.996511941391464005015480074215, −7.04897871578482242221733531225, −5.47234245773900780868499643101, −4.77456911625639953030195984093, −4.12639593438956303700358775353, −2.69353698661738615829747780950, −0.35310091079115328176212585526,
0.35310091079115328176212585526, 2.69353698661738615829747780950, 4.12639593438956303700358775353, 4.77456911625639953030195984093, 5.47234245773900780868499643101, 7.04897871578482242221733531225, 7.996511941391464005015480074215, 8.425450392327746416747695422336, 9.992054974021436185338001228082, 10.62854398650478205156915559147
