| L(s) = 1 | + 2.17·2-s + 2.70·4-s − 0.829·7-s + 1.53·8-s − 2.53·11-s − 4.24·13-s − 1.80·14-s − 2.07·16-s + 1.36·17-s − 19-s − 5.51·22-s + 3.36·23-s − 9.21·26-s − 2.24·28-s + 3.80·29-s − 0.290·31-s − 7.58·32-s + 2.97·34-s − 7.51·37-s − 2.17·38-s − 10.1·41-s − 5.35·43-s − 6.87·44-s + 7.31·46-s + 8.63·47-s − 6.31·49-s − 11.5·52-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 1.35·4-s − 0.313·7-s + 0.544·8-s − 0.765·11-s − 1.17·13-s − 0.481·14-s − 0.519·16-s + 0.332·17-s − 0.229·19-s − 1.17·22-s + 0.702·23-s − 1.80·26-s − 0.424·28-s + 0.705·29-s − 0.0522·31-s − 1.34·32-s + 0.509·34-s − 1.23·37-s − 0.352·38-s − 1.58·41-s − 0.816·43-s − 1.03·44-s + 1.07·46-s + 1.25·47-s − 0.901·49-s − 1.59·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 7 | \( 1 + 0.829T + 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 1.36T + 17T^{2} \) |
| 23 | \( 1 - 3.36T + 23T^{2} \) |
| 29 | \( 1 - 3.80T + 29T^{2} \) |
| 31 | \( 1 + 0.290T + 31T^{2} \) |
| 37 | \( 1 + 7.51T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 5.35T + 43T^{2} \) |
| 47 | \( 1 - 8.63T + 47T^{2} \) |
| 53 | \( 1 + 1.86T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 7.86T + 61T^{2} \) |
| 67 | \( 1 + 4.15T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 3.57T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−7.77822744787645235089896168053, −7.01024443557412263429559966803, −6.46860652691865058950365814678, −5.51596549916483381952026336983, −5.01285576592359694123292050262, −4.42278050745059800317415249350, −3.30682491076669451722999706880, −2.87847402303657117262150554461, −1.86649503685337924607749483943, 0,
1.86649503685337924607749483943, 2.87847402303657117262150554461, 3.30682491076669451722999706880, 4.42278050745059800317415249350, 5.01285576592359694123292050262, 5.51596549916483381952026336983, 6.46860652691865058950365814678, 7.01024443557412263429559966803, 7.77822744787645235089896168053
