| L(s) = 1 | + 2.69·2-s + 5.23·4-s − 0.608·5-s − 1.72·7-s + 8.70·8-s − 1.63·10-s + 1.33·11-s + 3.67·13-s − 4.64·14-s + 12.9·16-s + 2.76·17-s − 2.56·19-s − 3.18·20-s + 3.59·22-s + 0.539·23-s − 4.62·25-s + 9.87·26-s − 9.04·28-s + 8.13·29-s + 17.4·32-s + 7.43·34-s + 1.05·35-s + 7.74·37-s − 6.90·38-s − 5.29·40-s + 0.104·41-s − 3.00·43-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.61·4-s − 0.272·5-s − 0.652·7-s + 3.07·8-s − 0.517·10-s + 0.403·11-s + 1.01·13-s − 1.24·14-s + 3.23·16-s + 0.670·17-s − 0.589·19-s − 0.712·20-s + 0.767·22-s + 0.112·23-s − 0.925·25-s + 1.93·26-s − 1.70·28-s + 1.50·29-s + 3.08·32-s + 1.27·34-s + 0.177·35-s + 1.27·37-s − 1.12·38-s − 0.837·40-s + 0.0162·41-s − 0.458·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.754490468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.754490468\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 5 | \( 1 + 0.608T + 5T^{2} \) |
| 7 | \( 1 + 1.72T + 7T^{2} \) |
| 11 | \( 1 - 1.33T + 11T^{2} \) |
| 13 | \( 1 - 3.67T + 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 - 0.539T + 23T^{2} \) |
| 29 | \( 1 - 8.13T + 29T^{2} \) |
| 37 | \( 1 - 7.74T + 37T^{2} \) |
| 41 | \( 1 - 0.104T + 41T^{2} \) |
| 43 | \( 1 + 3.00T + 43T^{2} \) |
| 47 | \( 1 - 6.72T + 47T^{2} \) |
| 53 | \( 1 + 2.79T + 53T^{2} \) |
| 59 | \( 1 + 0.466T + 59T^{2} \) |
| 61 | \( 1 + 5.11T + 61T^{2} \) |
| 67 | \( 1 - 8.29T + 67T^{2} \) |
| 71 | \( 1 + 4.75T + 71T^{2} \) |
| 73 | \( 1 - 7.50T + 73T^{2} \) |
| 79 | \( 1 + 9.69T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 1.27T + 97T^{2} \) |
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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
−7.57376835977513684666828987869, −6.56180196578195673446683675989, −6.35420117422192291247548111145, −5.72727848049363553716598543718, −4.86769717305337098499369844387, −4.18235345812863194790622880090, −3.60643739332320597464382878212, −3.02877786956919783116219969174, −2.14739482274230771236770595273, −1.06570151396565971117416157193,
1.06570151396565971117416157193, 2.14739482274230771236770595273, 3.02877786956919783116219969174, 3.60643739332320597464382878212, 4.18235345812863194790622880090, 4.86769717305337098499369844387, 5.72727848049363553716598543718, 6.35420117422192291247548111145, 6.56180196578195673446683675989, 7.57376835977513684666828987869
