| L(s) = 1 | + 2-s + 1.25·3-s + 4-s − 0.642·5-s + 1.25·6-s + 4.42·7-s + 8-s − 1.42·9-s − 0.642·10-s − 3.47·11-s + 1.25·12-s + 6.97·13-s + 4.42·14-s − 0.807·15-s + 16-s + 5.11·17-s − 1.42·18-s − 3.88·19-s − 0.642·20-s + 5.56·21-s − 3.47·22-s + 3.72·23-s + 1.25·24-s − 4.58·25-s + 6.97·26-s − 5.55·27-s + 4.42·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.725·3-s + 0.5·4-s − 0.287·5-s + 0.513·6-s + 1.67·7-s + 0.353·8-s − 0.473·9-s − 0.203·10-s − 1.04·11-s + 0.362·12-s + 1.93·13-s + 1.18·14-s − 0.208·15-s + 0.250·16-s + 1.24·17-s − 0.334·18-s − 0.891·19-s − 0.143·20-s + 1.21·21-s − 0.740·22-s + 0.777·23-s + 0.256·24-s − 0.917·25-s + 1.36·26-s − 1.06·27-s + 0.836·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.036870102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.036870102\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 1.25T + 3T^{2} \) |
| 5 | \( 1 + 0.642T + 5T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 - 6.97T + 13T^{2} \) |
| 17 | \( 1 - 5.11T + 17T^{2} \) |
| 19 | \( 1 + 3.88T + 19T^{2} \) |
| 23 | \( 1 - 3.72T + 23T^{2} \) |
| 29 | \( 1 - 6.35T + 29T^{2} \) |
| 37 | \( 1 - 6.42T + 37T^{2} \) |
| 41 | \( 1 + 4.86T + 41T^{2} \) |
| 43 | \( 1 - 3.54T + 43T^{2} \) |
| 47 | \( 1 + 6.37T + 47T^{2} \) |
| 53 | \( 1 + 6.87T + 53T^{2} \) |
| 59 | \( 1 - 9.54T + 59T^{2} \) |
| 61 | \( 1 + 1.88T + 61T^{2} \) |
| 67 | \( 1 + 8.46T + 67T^{2} \) |
| 71 | \( 1 + 3.32T + 71T^{2} \) |
| 73 | \( 1 + 1.61T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 - 8.31T + 83T^{2} \) |
| 89 | \( 1 - 8.25T + 89T^{2} \) |
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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
−7.953517469391018633577935197205, −7.83051547344227379560156626336, −6.53064828436589115602903286089, −5.78031546268684133482233720160, −5.16641694746576368668535862452, −4.40747864822713939891106886629, −3.62295143392071345340567381288, −2.91611827038487677005155166980, −2.01494430080248753861066644879, −1.10896277170016599398232873142,
1.10896277170016599398232873142, 2.01494430080248753861066644879, 2.91611827038487677005155166980, 3.62295143392071345340567381288, 4.40747864822713939891106886629, 5.16641694746576368668535862452, 5.78031546268684133482233720160, 6.53064828436589115602903286089, 7.83051547344227379560156626336, 7.953517469391018633577935197205
