| L(s) = 1 | + 2-s − 3-s + 4-s + 0.724·5-s − 6-s − 0.160·7-s + 8-s + 9-s + 0.724·10-s + 4.82·11-s − 12-s − 5.08·13-s − 0.160·14-s − 0.724·15-s + 16-s − 17-s + 18-s + 0.132·19-s + 0.724·20-s + 0.160·21-s + 4.82·22-s − 6.42·23-s − 24-s − 4.47·25-s − 5.08·26-s − 27-s − 0.160·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.323·5-s − 0.408·6-s − 0.0606·7-s + 0.353·8-s + 0.333·9-s + 0.228·10-s + 1.45·11-s − 0.288·12-s − 1.41·13-s − 0.0428·14-s − 0.186·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.0304·19-s + 0.161·20-s + 0.0350·21-s + 1.02·22-s − 1.33·23-s − 0.204·24-s − 0.895·25-s − 0.997·26-s − 0.192·27-s − 0.0303·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 0.724T + 5T^{2} \) |
| 7 | \( 1 + 0.160T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 5.08T + 13T^{2} \) |
| 19 | \( 1 - 0.132T + 19T^{2} \) |
| 23 | \( 1 + 6.42T + 23T^{2} \) |
| 29 | \( 1 + 2.70T + 29T^{2} \) |
| 31 | \( 1 + 1.82T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 9.83T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 3.55T + 53T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 + 0.809T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 1.36T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 2.50T + 83T^{2} \) |
| 89 | \( 1 - 7.70T + 89T^{2} \) |
| 97 | \( 1 + 1.86T + 97T^{2} \) |
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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
−7.54056516431011632016283421925, −6.67877397645668787457128452090, −6.38644504630603813168438338389, −5.51632601199810195011670674923, −4.88506762076866003636520502472, −4.11154241007131427871083125584, −3.45929356379128724664942331574, −2.22622030437138064189540243038, −1.57117136663077017713117899162, 0,
1.57117136663077017713117899162, 2.22622030437138064189540243038, 3.45929356379128724664942331574, 4.11154241007131427871083125584, 4.88506762076866003636520502472, 5.51632601199810195011670674923, 6.38644504630603813168438338389, 6.67877397645668787457128452090, 7.54056516431011632016283421925
