| L(s) = 1 | − 2-s + 0.381·3-s + 4-s − 0.381·6-s + 2.61·7-s − 8-s − 2.85·9-s + 2·11-s + 0.381·12-s + 0.381·13-s − 2.61·14-s + 16-s + 3.38·17-s + 2.85·18-s − 2·19-s + 21-s − 2·22-s + 4.85·23-s − 0.381·24-s − 0.381·26-s − 2.23·27-s + 2.61·28-s − 29-s − 3.85·31-s − 32-s + 0.763·33-s − 3.38·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.220·3-s + 0.5·4-s − 0.155·6-s + 0.989·7-s − 0.353·8-s − 0.951·9-s + 0.603·11-s + 0.110·12-s + 0.105·13-s − 0.699·14-s + 0.250·16-s + 0.820·17-s + 0.672·18-s − 0.458·19-s + 0.218·21-s − 0.426·22-s + 1.01·23-s − 0.0779·24-s − 0.0749·26-s − 0.430·27-s + 0.494·28-s − 0.185·29-s − 0.692·31-s − 0.176·32-s + 0.132·33-s − 0.580·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.451356589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.451356589\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 0.381T + 3T^{2} \) |
| 7 | \( 1 - 2.61T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 0.381T + 13T^{2} \) |
| 17 | \( 1 - 3.38T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 4.85T + 23T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 - 8.94T + 37T^{2} \) |
| 41 | \( 1 - 7.70T + 41T^{2} \) |
| 43 | \( 1 - 3.38T + 43T^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 - 9.38T + 53T^{2} \) |
| 59 | \( 1 - 2.85T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 8.94T + 67T^{2} \) |
| 71 | \( 1 + 5.70T + 71T^{2} \) |
| 73 | \( 1 + 5.56T + 73T^{2} \) |
| 79 | \( 1 - 6.56T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 7.14T + 97T^{2} \) |
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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
−9.153627672888734799519349206160, −8.919336176833270735228795466362, −7.904701529681630813566634952368, −7.46545746142862133611085481301, −6.25765777148109520918391865725, −5.53960776535579606300237729488, −4.43864202689874915728384500504, −3.26132254108796269796577948279, −2.20251622547074173310977380191, −0.991612327571351067344683468349,
0.991612327571351067344683468349, 2.20251622547074173310977380191, 3.26132254108796269796577948279, 4.43864202689874915728384500504, 5.53960776535579606300237729488, 6.25765777148109520918391865725, 7.46545746142862133611085481301, 7.904701529681630813566634952368, 8.919336176833270735228795466362, 9.153627672888734799519349206160
