| L(s) = 1 | + 1.76·2-s + 3-s + 1.12·4-s + 5-s + 1.76·6-s + 1.48·7-s − 1.54·8-s + 9-s + 1.76·10-s + 2.88·11-s + 1.12·12-s − 2.36·13-s + 2.62·14-s + 15-s − 4.98·16-s + 4.73·17-s + 1.76·18-s − 2.96·19-s + 1.12·20-s + 1.48·21-s + 5.10·22-s + 1.36·23-s − 1.54·24-s + 25-s − 4.17·26-s + 27-s + 1.67·28-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.577·3-s + 0.563·4-s + 0.447·5-s + 0.721·6-s + 0.561·7-s − 0.546·8-s + 0.333·9-s + 0.559·10-s + 0.870·11-s + 0.325·12-s − 0.655·13-s + 0.702·14-s + 0.258·15-s − 1.24·16-s + 1.14·17-s + 0.416·18-s − 0.681·19-s + 0.251·20-s + 0.324·21-s + 1.08·22-s + 0.284·23-s − 0.315·24-s + 0.200·25-s − 0.819·26-s + 0.192·27-s + 0.316·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.700465887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.700465887\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 1.76T + 2T^{2} \) |
| 7 | \( 1 - 1.48T + 7T^{2} \) |
| 11 | \( 1 - 2.88T + 11T^{2} \) |
| 13 | \( 1 + 2.36T + 13T^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 19 | \( 1 + 2.96T + 19T^{2} \) |
| 23 | \( 1 - 1.36T + 23T^{2} \) |
| 29 | \( 1 - 5.35T + 29T^{2} \) |
| 31 | \( 1 + 0.0834T + 31T^{2} \) |
| 37 | \( 1 - 7.83T + 37T^{2} \) |
| 41 | \( 1 - 6.72T + 41T^{2} \) |
| 43 | \( 1 - 9.96T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 5.51T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 6.34T + 67T^{2} \) |
| 71 | \( 1 + 9.23T + 71T^{2} \) |
| 73 | \( 1 - 7.49T + 73T^{2} \) |
| 79 | \( 1 + 4.72T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 0.833T + 97T^{2} \) |
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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
−7.974161948750216249937510954953, −7.31256863793164455937626159847, −6.36282408520644205967173140821, −5.94039992165060280798693160339, −4.96169463197847370427549309167, −4.50707587367887704122388465935, −3.74334916010063860980820216935, −2.90994290098727883549542378142, −2.21649072025772513371592213773, −1.07284269330874549571011250688,
1.07284269330874549571011250688, 2.21649072025772513371592213773, 2.90994290098727883549542378142, 3.74334916010063860980820216935, 4.50707587367887704122388465935, 4.96169463197847370427549309167, 5.94039992165060280798693160339, 6.36282408520644205967173140821, 7.31256863793164455937626159847, 7.974161948750216249937510954953
