| L(s) = 1 | + 3-s − 2.48·5-s + 0.647·7-s + 9-s + 6.04·11-s − 4.46·13-s − 2.48·15-s + 6.16·17-s − 4.37·19-s + 0.647·21-s − 0.856·23-s + 1.18·25-s + 27-s − 0.764·29-s + 2.34·31-s + 6.04·33-s − 1.60·35-s + 2.94·37-s − 4.46·39-s + 4.61·41-s + 8.12·43-s − 2.48·45-s − 5.60·47-s − 6.58·49-s + 6.16·51-s − 5.61·53-s − 15.0·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.11·5-s + 0.244·7-s + 0.333·9-s + 1.82·11-s − 1.23·13-s − 0.642·15-s + 1.49·17-s − 1.00·19-s + 0.141·21-s − 0.178·23-s + 0.236·25-s + 0.192·27-s − 0.141·29-s + 0.421·31-s + 1.05·33-s − 0.272·35-s + 0.484·37-s − 0.714·39-s + 0.721·41-s + 1.23·43-s − 0.370·45-s − 0.817·47-s − 0.940·49-s + 0.863·51-s − 0.771·53-s − 2.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.166153446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.166153446\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 + 2.48T + 5T^{2} \) |
| 7 | \( 1 - 0.647T + 7T^{2} \) |
| 11 | \( 1 - 6.04T + 11T^{2} \) |
| 13 | \( 1 + 4.46T + 13T^{2} \) |
| 17 | \( 1 - 6.16T + 17T^{2} \) |
| 19 | \( 1 + 4.37T + 19T^{2} \) |
| 23 | \( 1 + 0.856T + 23T^{2} \) |
| 29 | \( 1 + 0.764T + 29T^{2} \) |
| 31 | \( 1 - 2.34T + 31T^{2} \) |
| 37 | \( 1 - 2.94T + 37T^{2} \) |
| 41 | \( 1 - 4.61T + 41T^{2} \) |
| 43 | \( 1 - 8.12T + 43T^{2} \) |
| 47 | \( 1 + 5.60T + 47T^{2} \) |
| 53 | \( 1 + 5.61T + 53T^{2} \) |
| 59 | \( 1 - 6.37T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 + 0.458T + 67T^{2} \) |
| 71 | \( 1 + 5.77T + 71T^{2} \) |
| 73 | \( 1 - 6.72T + 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 2.95T + 97T^{2} \) |
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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
−8.017640660100094467384907645592, −7.51977511532422299872834928833, −6.82910467277803944307586489937, −6.08203072245582976227478303848, −5.00098548642997197392155288853, −4.17443635579422421612350920058, −3.81060578761654082436217947068, −2.90220982180101041038398271048, −1.85208025600190864545755252769, −0.76940420664926547619483506838,
0.76940420664926547619483506838, 1.85208025600190864545755252769, 2.90220982180101041038398271048, 3.81060578761654082436217947068, 4.17443635579422421612350920058, 5.00098548642997197392155288853, 6.08203072245582976227478303848, 6.82910467277803944307586489937, 7.51977511532422299872834928833, 8.017640660100094467384907645592
