| L(s) = 1 | + 0.134·2-s + 3-s − 1.98·4-s + 5-s + 0.134·6-s + 1.40·7-s − 0.534·8-s + 9-s + 0.134·10-s − 1.28·11-s − 1.98·12-s + 3.37·13-s + 0.188·14-s + 15-s + 3.89·16-s + 0.861·17-s + 0.134·18-s + 4.53·19-s − 1.98·20-s + 1.40·21-s − 0.172·22-s − 0.534·24-s + 25-s + 0.453·26-s + 27-s − 2.79·28-s + 0.141·29-s + ⋯ |
| L(s) = 1 | + 0.0948·2-s + 0.577·3-s − 0.991·4-s + 0.447·5-s + 0.0547·6-s + 0.532·7-s − 0.188·8-s + 0.333·9-s + 0.0424·10-s − 0.388·11-s − 0.572·12-s + 0.937·13-s + 0.0504·14-s + 0.258·15-s + 0.973·16-s + 0.209·17-s + 0.0316·18-s + 1.04·19-s − 0.443·20-s + 0.307·21-s − 0.0368·22-s − 0.109·24-s + 0.200·25-s + 0.0888·26-s + 0.192·27-s − 0.527·28-s + 0.0263·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.727947787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.727947787\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.134T + 2T^{2} \) |
| 7 | \( 1 - 1.40T + 7T^{2} \) |
| 11 | \( 1 + 1.28T + 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 - 0.861T + 17T^{2} \) |
| 19 | \( 1 - 4.53T + 19T^{2} \) |
| 29 | \( 1 - 0.141T + 29T^{2} \) |
| 31 | \( 1 - 2.54T + 31T^{2} \) |
| 37 | \( 1 - 1.11T + 37T^{2} \) |
| 41 | \( 1 + 6.00T + 41T^{2} \) |
| 43 | \( 1 - 0.457T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 - 6.86T + 53T^{2} \) |
| 59 | \( 1 + 2.57T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 1.94T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 9.02T + 79T^{2} \) |
| 83 | \( 1 - 8.77T + 83T^{2} \) |
| 89 | \( 1 - 4.17T + 89T^{2} \) |
| 97 | \( 1 + 9.81T + 97T^{2} \) |
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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
−8.017543494658908997863972861615, −7.35331258776388870090603747342, −6.36488182251117091140046536120, −5.59120579117895246561514671424, −5.04652478019744869724105564813, −4.28516493110937257811771703361, −3.52385172479563054554301578949, −2.82832610138016593424210893792, −1.68655640252525224516858971259, −0.844932698520024596474494836535,
0.844932698520024596474494836535, 1.68655640252525224516858971259, 2.82832610138016593424210893792, 3.52385172479563054554301578949, 4.28516493110937257811771703361, 5.04652478019744869724105564813, 5.59120579117895246561514671424, 6.36488182251117091140046536120, 7.35331258776388870090603747342, 8.017543494658908997863972861615
