| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 6.42·11-s − 13-s − 14-s + 16-s + 5.42·17-s + 8.42·19-s − 6.42·22-s + 3.42·23-s + 26-s + 28-s − 3.42·29-s − 1.42·31-s − 32-s − 5.42·34-s + 8.84·37-s − 8.42·38-s + 10.4·41-s + 7·43-s + 6.42·44-s − 3.42·46-s − 10.4·47-s + 49-s − 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.377·7-s − 0.353·8-s + 1.93·11-s − 0.277·13-s − 0.267·14-s + 0.250·16-s + 1.31·17-s + 1.93·19-s − 1.36·22-s + 0.714·23-s + 0.196·26-s + 0.188·28-s − 0.635·29-s − 0.255·31-s − 0.176·32-s − 0.930·34-s + 1.45·37-s − 1.36·38-s + 1.62·41-s + 1.06·43-s + 0.968·44-s − 0.504·46-s − 1.52·47-s + 0.142·49-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.302700770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.302700770\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 6.42T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 5.42T + 17T^{2} \) |
| 19 | \( 1 - 8.42T + 19T^{2} \) |
| 23 | \( 1 - 3.42T + 23T^{2} \) |
| 29 | \( 1 + 3.42T + 29T^{2} \) |
| 31 | \( 1 + 1.42T + 31T^{2} \) |
| 37 | \( 1 - 8.84T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 7T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 5T + 53T^{2} \) |
| 59 | \( 1 - 7.42T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 3T + 67T^{2} \) |
| 71 | \( 1 - 5.42T + 71T^{2} \) |
| 73 | \( 1 + 2.42T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 7.57T + 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−7.55328977730697777566663347456, −7.30747627034380874196053803739, −6.42508496443011791599354783734, −5.74662113277053634053109974186, −5.07019661601460422570935549524, −4.03948931634693608489759938734, −3.42169317763331040942022624053, −2.53580149543667072576251740225, −1.28035722352257182438114929080, −1.01260825345890657038299204786,
1.01260825345890657038299204786, 1.28035722352257182438114929080, 2.53580149543667072576251740225, 3.42169317763331040942022624053, 4.03948931634693608489759938734, 5.07019661601460422570935549524, 5.74662113277053634053109974186, 6.42508496443011791599354783734, 7.30747627034380874196053803739, 7.55328977730697777566663347456
