| L(s) = 1 | + 2-s + 4-s + 2.89·5-s + 2.77·7-s + 8-s + 2.89·10-s + 5.45·11-s + 0.365·13-s + 2.77·14-s + 16-s − 6.29·17-s − 6.13·19-s + 2.89·20-s + 5.45·22-s + 3.37·25-s + 0.365·26-s + 2.77·28-s − 3.41·29-s + 7.68·31-s + 32-s − 6.29·34-s + 8.01·35-s − 4.82·37-s − 6.13·38-s + 2.89·40-s + 4.23·41-s + 7.03·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.29·5-s + 1.04·7-s + 0.353·8-s + 0.915·10-s + 1.64·11-s + 0.101·13-s + 0.740·14-s + 0.250·16-s − 1.52·17-s − 1.40·19-s + 0.647·20-s + 1.16·22-s + 0.674·25-s + 0.0717·26-s + 0.523·28-s − 0.633·29-s + 1.37·31-s + 0.176·32-s − 1.08·34-s + 1.35·35-s − 0.793·37-s − 0.995·38-s + 0.457·40-s + 0.661·41-s + 1.07·43-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
| \(L(1)\) |
\(\approx\) |
\(5.612603071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.612603071\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.89T + 5T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 11 | \( 1 - 5.45T + 11T^{2} \) |
| 13 | \( 1 - 0.365T + 13T^{2} \) |
| 17 | \( 1 + 6.29T + 17T^{2} \) |
| 19 | \( 1 + 6.13T + 19T^{2} \) |
| 29 | \( 1 + 3.41T + 29T^{2} \) |
| 31 | \( 1 - 7.68T + 31T^{2} \) |
| 37 | \( 1 + 4.82T + 37T^{2} \) |
| 41 | \( 1 - 4.23T + 41T^{2} \) |
| 43 | \( 1 - 7.03T + 43T^{2} \) |
| 47 | \( 1 - 6.68T + 47T^{2} \) |
| 53 | \( 1 - 7.13T + 53T^{2} \) |
| 59 | \( 1 + 0.440T + 59T^{2} \) |
| 61 | \( 1 + 8.13T + 61T^{2} \) |
| 67 | \( 1 + 4.44T + 67T^{2} \) |
| 71 | \( 1 - 1.41T + 71T^{2} \) |
| 73 | \( 1 + 3.94T + 73T^{2} \) |
| 79 | \( 1 - 3.97T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 4.33T + 97T^{2} \) |
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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
−7.50582386404146829697435627843, −6.72995187078289509668826285074, −6.19966315803198631000833504904, −5.82838476027493483471626592155, −4.68325376380762746428009780281, −4.46448749549165899991233787201, −3.61018474491636678940978575570, −2.23666385064384785127449698962, −2.06395551642873216207824455572, −1.08851538721877189090992021306,
1.08851538721877189090992021306, 2.06395551642873216207824455572, 2.23666385064384785127449698962, 3.61018474491636678940978575570, 4.46448749549165899991233787201, 4.68325376380762746428009780281, 5.82838476027493483471626592155, 6.19966315803198631000833504904, 6.72995187078289509668826285074, 7.50582386404146829697435627843
