| L(s) = 1 | − 3.82·2-s + 3·3-s + 6.65·4-s − 5·5-s − 11.4·6-s + 5.14·8-s + 9·9-s + 19.1·10-s + 48.5·11-s + 19.9·12-s + 43.6·13-s − 15·15-s − 72.9·16-s + 67.6·17-s − 34.4·18-s + 93.2·19-s − 33.2·20-s − 185.·22-s − 104.·23-s + 15.4·24-s + 25·25-s − 167.·26-s + 27·27-s − 58.7·29-s + 57.4·30-s + 9.08·31-s + 238.·32-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.577·3-s + 0.832·4-s − 0.447·5-s − 0.781·6-s + 0.227·8-s + 0.333·9-s + 0.605·10-s + 1.33·11-s + 0.480·12-s + 0.931·13-s − 0.258·15-s − 1.13·16-s + 0.965·17-s − 0.451·18-s + 1.12·19-s − 0.372·20-s − 1.80·22-s − 0.944·23-s + 0.131·24-s + 0.200·25-s − 1.26·26-s + 0.192·27-s − 0.376·29-s + 0.349·30-s + 0.0526·31-s + 1.31·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.335480330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.335480330\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.82T + 8T^{2} \) |
| 11 | \( 1 - 48.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 43.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 93.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 58.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 9.08T + 2.97e4T^{2} \) |
| 37 | \( 1 + 252.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 276.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 92.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 582.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 623.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 524.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 352.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 736.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 492.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 872.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 529.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 385.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 463.T + 9.12e5T^{2} \) |
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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
−9.888592815507846826580610397486, −8.888912747320645013370379942907, −8.587230537100032457373585557347, −7.55439853012923712801600649034, −6.99363800474875261165153274785, −5.72784522162088994712727506113, −4.18368211684997589417324072063, −3.37405971014763555008885041208, −1.72816161663802529250728229333, −0.851499036191040435296454975892,
0.851499036191040435296454975892, 1.72816161663802529250728229333, 3.37405971014763555008885041208, 4.18368211684997589417324072063, 5.72784522162088994712727506113, 6.99363800474875261165153274785, 7.55439853012923712801600649034, 8.587230537100032457373585557347, 8.888912747320645013370379942907, 9.888592815507846826580610397486
