| L(s) = 1 | − 2-s + 4-s + 2.34·5-s + 3.57·7-s − 8-s − 2.34·10-s − 2.71·11-s + 5.41·13-s − 3.57·14-s + 16-s + 3.87·17-s + 2.34·20-s + 2.71·22-s + 8.23·23-s + 0.509·25-s − 5.41·26-s + 3.57·28-s − 3.53·29-s + 6.53·31-s − 32-s − 3.87·34-s + 8.38·35-s + 0.389·37-s − 2.34·40-s + 1.94·41-s + 5.02·43-s − 2.71·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.04·5-s + 1.35·7-s − 0.353·8-s − 0.742·10-s − 0.819·11-s + 1.50·13-s − 0.955·14-s + 0.250·16-s + 0.940·17-s + 0.524·20-s + 0.579·22-s + 1.71·23-s + 0.101·25-s − 1.06·26-s + 0.675·28-s − 0.655·29-s + 1.17·31-s − 0.176·32-s − 0.665·34-s + 1.41·35-s + 0.0639·37-s − 0.371·40-s + 0.303·41-s + 0.765·43-s − 0.409·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.522671434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.522671434\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 2.34T + 5T^{2} \) |
| 7 | \( 1 - 3.57T + 7T^{2} \) |
| 11 | \( 1 + 2.71T + 11T^{2} \) |
| 13 | \( 1 - 5.41T + 13T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 23 | \( 1 - 8.23T + 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 - 6.53T + 31T^{2} \) |
| 37 | \( 1 - 0.389T + 37T^{2} \) |
| 41 | \( 1 - 1.94T + 41T^{2} \) |
| 43 | \( 1 - 5.02T + 43T^{2} \) |
| 47 | \( 1 + 2.98T + 47T^{2} \) |
| 53 | \( 1 + 8.30T + 53T^{2} \) |
| 59 | \( 1 + 2.73T + 59T^{2} \) |
| 61 | \( 1 - 6.29T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 9.02T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 8.17T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 8.60T + 97T^{2} \) |
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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
−8.016826867083664141647503558455, −7.58983506707350831701144926919, −6.59875512573771911672675685108, −5.87156405001760402952061064560, −5.33498883020790159648344141025, −4.59277340792693740992279782633, −3.38107225132242453385052304973, −2.52993227137091815159084671529, −1.57614478462379052971758104879, −1.03867185892129890281571099412,
1.03867185892129890281571099412, 1.57614478462379052971758104879, 2.52993227137091815159084671529, 3.38107225132242453385052304973, 4.59277340792693740992279782633, 5.33498883020790159648344141025, 5.87156405001760402952061064560, 6.59875512573771911672675685108, 7.58983506707350831701144926919, 8.016826867083664141647503558455
