| L(s) = 1 | − 2.70·2-s − 3-s + 5.34·4-s + 5-s + 2.70·6-s + 0.462·7-s − 9.05·8-s + 9-s − 2.70·10-s + 2.98·11-s − 5.34·12-s − 3.67·13-s − 1.25·14-s − 15-s + 13.8·16-s + 6.00·17-s − 2.70·18-s + 1.58·19-s + 5.34·20-s − 0.462·21-s − 8.08·22-s + 3.06·23-s + 9.05·24-s + 25-s + 9.97·26-s − 27-s + 2.46·28-s + ⋯ |
| L(s) = 1 | − 1.91·2-s − 0.577·3-s + 2.67·4-s + 0.447·5-s + 1.10·6-s + 0.174·7-s − 3.19·8-s + 0.333·9-s − 0.856·10-s + 0.899·11-s − 1.54·12-s − 1.02·13-s − 0.334·14-s − 0.258·15-s + 3.45·16-s + 1.45·17-s − 0.638·18-s + 0.362·19-s + 1.19·20-s − 0.100·21-s − 1.72·22-s + 0.638·23-s + 1.84·24-s + 0.200·25-s + 1.95·26-s − 0.192·27-s + 0.466·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9172018929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9172018929\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 7 | \( 1 - 0.462T + 7T^{2} \) |
| 11 | \( 1 - 2.98T + 11T^{2} \) |
| 13 | \( 1 + 3.67T + 13T^{2} \) |
| 17 | \( 1 - 6.00T + 17T^{2} \) |
| 19 | \( 1 - 1.58T + 19T^{2} \) |
| 23 | \( 1 - 3.06T + 23T^{2} \) |
| 29 | \( 1 - 6.10T + 29T^{2} \) |
| 31 | \( 1 - 9.25T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 8.97T + 41T^{2} \) |
| 43 | \( 1 + 9.18T + 43T^{2} \) |
| 47 | \( 1 + 6.45T + 47T^{2} \) |
| 53 | \( 1 - 6.24T + 53T^{2} \) |
| 59 | \( 1 - 0.300T + 59T^{2} \) |
| 61 | \( 1 + 1.63T + 61T^{2} \) |
| 67 | \( 1 + 1.57T + 67T^{2} \) |
| 71 | \( 1 - 2.94T + 71T^{2} \) |
| 73 | \( 1 - 4.30T + 73T^{2} \) |
| 79 | \( 1 + 0.601T + 79T^{2} \) |
| 83 | \( 1 - 9.53T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−7.957387219929890579048859595763, −7.68336688325402187525112672032, −6.64926780897968995776080288291, −6.42819173264518218486254114220, −5.50028597670006685657725217989, −4.61651810610707871131382135009, −3.18312628227949103970487311363, −2.44929805660369254424860546171, −1.31083984124263169788486869160, −0.804961450723042922175364764047,
0.804961450723042922175364764047, 1.31083984124263169788486869160, 2.44929805660369254424860546171, 3.18312628227949103970487311363, 4.61651810610707871131382135009, 5.50028597670006685657725217989, 6.42819173264518218486254114220, 6.64926780897968995776080288291, 7.68336688325402187525112672032, 7.957387219929890579048859595763
