| L(s) = 1 | + 2-s + 4-s − 2.43·5-s − 1.85·7-s + 8-s − 2.43·10-s + 5.55·11-s + 1.14·13-s − 1.85·14-s + 16-s + 7.50·17-s + 4.47·19-s − 2.43·20-s + 5.55·22-s + 0.916·25-s + 1.14·26-s − 1.85·28-s − 2.93·29-s − 2.07·31-s + 32-s + 7.50·34-s + 4.51·35-s − 7.73·37-s + 4.47·38-s − 2.43·40-s − 0.688·41-s + 8.56·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.08·5-s − 0.701·7-s + 0.353·8-s − 0.769·10-s + 1.67·11-s + 0.318·13-s − 0.496·14-s + 0.250·16-s + 1.82·17-s + 1.02·19-s − 0.543·20-s + 1.18·22-s + 0.183·25-s + 0.225·26-s − 0.350·28-s − 0.545·29-s − 0.373·31-s + 0.176·32-s + 1.28·34-s + 0.763·35-s − 1.27·37-s + 0.726·38-s − 0.384·40-s − 0.107·41-s + 1.30·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.018583830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.018583830\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2.43T + 5T^{2} \) |
| 7 | \( 1 + 1.85T + 7T^{2} \) |
| 11 | \( 1 - 5.55T + 11T^{2} \) |
| 13 | \( 1 - 1.14T + 13T^{2} \) |
| 17 | \( 1 - 7.50T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 29 | \( 1 + 2.93T + 29T^{2} \) |
| 31 | \( 1 + 2.07T + 31T^{2} \) |
| 37 | \( 1 + 7.73T + 37T^{2} \) |
| 41 | \( 1 + 0.688T + 41T^{2} \) |
| 43 | \( 1 - 8.56T + 43T^{2} \) |
| 47 | \( 1 + 7.09T + 47T^{2} \) |
| 53 | \( 1 - 5.38T + 53T^{2} \) |
| 59 | \( 1 - 4.03T + 59T^{2} \) |
| 61 | \( 1 - 0.379T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 9.94T + 97T^{2} \) |
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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
−7.46970875367416503644658142792, −7.07316247058605138885057518394, −6.26413389864884813232742076633, −5.67404191974714043334403113804, −4.89340535580347058150155210756, −3.83538978493578987662843313570, −3.65518067191404512207765126205, −3.08058503537318925720027895249, −1.66821006098613255310472554465, −0.78532581877856188028387648433,
0.78532581877856188028387648433, 1.66821006098613255310472554465, 3.08058503537318925720027895249, 3.65518067191404512207765126205, 3.83538978493578987662843313570, 4.89340535580347058150155210756, 5.67404191974714043334403113804, 6.26413389864884813232742076633, 7.07316247058605138885057518394, 7.46970875367416503644658142792
