| L(s) = 1 | + 1.63·2-s + 0.663·4-s + 2.08·5-s − 1.57·7-s − 2.18·8-s + 3.40·10-s + 0.321·11-s + 4.63·13-s − 2.56·14-s − 4.88·16-s − 3.48·17-s + 19-s + 1.38·20-s + 0.524·22-s − 1.40·23-s − 0.656·25-s + 7.56·26-s − 1.04·28-s + 4.67·29-s + 7.83·31-s − 3.61·32-s − 5.69·34-s − 3.27·35-s − 7.80·37-s + 1.63·38-s − 4.54·40-s + 7.94·41-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 0.331·4-s + 0.932·5-s − 0.594·7-s − 0.771·8-s + 1.07·10-s + 0.0969·11-s + 1.28·13-s − 0.686·14-s − 1.22·16-s − 0.846·17-s + 0.229·19-s + 0.309·20-s + 0.111·22-s − 0.293·23-s − 0.131·25-s + 1.48·26-s − 0.197·28-s + 0.867·29-s + 1.40·31-s − 0.638·32-s − 0.976·34-s − 0.554·35-s − 1.28·37-s + 0.264·38-s − 0.718·40-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.894889415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.894889415\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 1.63T + 2T^{2} \) |
| 5 | \( 1 - 2.08T + 5T^{2} \) |
| 7 | \( 1 + 1.57T + 7T^{2} \) |
| 11 | \( 1 - 0.321T + 11T^{2} \) |
| 13 | \( 1 - 4.63T + 13T^{2} \) |
| 17 | \( 1 + 3.48T + 17T^{2} \) |
| 23 | \( 1 + 1.40T + 23T^{2} \) |
| 29 | \( 1 - 4.67T + 29T^{2} \) |
| 31 | \( 1 - 7.83T + 31T^{2} \) |
| 37 | \( 1 + 7.80T + 37T^{2} \) |
| 41 | \( 1 - 7.94T + 41T^{2} \) |
| 43 | \( 1 - 8.29T + 43T^{2} \) |
| 53 | \( 1 - 6.82T + 53T^{2} \) |
| 59 | \( 1 + 1.85T + 59T^{2} \) |
| 61 | \( 1 - 1.33T + 61T^{2} \) |
| 67 | \( 1 - 4.60T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 5.09T + 73T^{2} \) |
| 79 | \( 1 + 4.36T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−7.75180092113894815334992291330, −6.51191668702310968329148495886, −6.43967703842132146287784289792, −5.75844457474002133873822281079, −5.06808254817678849903898900025, −4.24340092934122440572981246973, −3.65489089262140841484094928593, −2.81698095505084242523518395988, −2.09616339503298645166664193627, −0.817601707304884671168612751399,
0.817601707304884671168612751399, 2.09616339503298645166664193627, 2.81698095505084242523518395988, 3.65489089262140841484094928593, 4.24340092934122440572981246973, 5.06808254817678849903898900025, 5.75844457474002133873822281079, 6.43967703842132146287784289792, 6.51191668702310968329148495886, 7.75180092113894815334992291330
