| L(s) = 1 | − 1.10·2-s − 0.779·4-s + 5-s − 2.51·7-s + 3.07·8-s − 1.10·10-s + 11-s + 1.77·13-s + 2.78·14-s − 1.83·16-s + 1.71·17-s − 19-s − 0.779·20-s − 1.10·22-s − 6.14·23-s + 25-s − 1.96·26-s + 1.96·28-s − 8.27·29-s + 2.80·31-s − 4.11·32-s − 1.90·34-s − 2.51·35-s + 8.90·37-s + 1.10·38-s + 3.07·40-s + 4.57·41-s + ⋯ |
| L(s) = 1 | − 0.781·2-s − 0.389·4-s + 0.447·5-s − 0.951·7-s + 1.08·8-s − 0.349·10-s + 0.301·11-s + 0.492·13-s + 0.743·14-s − 0.458·16-s + 0.417·17-s − 0.229·19-s − 0.174·20-s − 0.235·22-s − 1.28·23-s + 0.200·25-s − 0.384·26-s + 0.370·28-s − 1.53·29-s + 0.502·31-s − 0.727·32-s − 0.325·34-s − 0.425·35-s + 1.46·37-s + 0.179·38-s + 0.485·40-s + 0.713·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.10T + 2T^{2} \) |
| 7 | \( 1 + 2.51T + 7T^{2} \) |
| 13 | \( 1 - 1.77T + 13T^{2} \) |
| 17 | \( 1 - 1.71T + 17T^{2} \) |
| 23 | \( 1 + 6.14T + 23T^{2} \) |
| 29 | \( 1 + 8.27T + 29T^{2} \) |
| 31 | \( 1 - 2.80T + 31T^{2} \) |
| 37 | \( 1 - 8.90T + 37T^{2} \) |
| 41 | \( 1 - 4.57T + 41T^{2} \) |
| 43 | \( 1 - 2.26T + 43T^{2} \) |
| 47 | \( 1 - 2.25T + 47T^{2} \) |
| 53 | \( 1 - 8.96T + 53T^{2} \) |
| 59 | \( 1 + 2.94T + 59T^{2} \) |
| 61 | \( 1 + 7.16T + 61T^{2} \) |
| 67 | \( 1 - 5.42T + 67T^{2} \) |
| 71 | \( 1 + 7.37T + 71T^{2} \) |
| 73 | \( 1 + 7.59T + 73T^{2} \) |
| 79 | \( 1 + 9.25T + 79T^{2} \) |
| 83 | \( 1 + 7.15T + 83T^{2} \) |
| 89 | \( 1 - 9.20T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−7.57502462689885957836701094215, −6.72787352156314946408273397122, −5.96695048575851939400112370573, −5.57567267625807772020119585173, −4.35704676794943702884110059884, −3.94823509521723399414541586006, −2.97529924310963523799684471790, −1.97516814421972413266331947365, −1.05614221404034277179471356342, 0,
1.05614221404034277179471356342, 1.97516814421972413266331947365, 2.97529924310963523799684471790, 3.94823509521723399414541586006, 4.35704676794943702884110059884, 5.57567267625807772020119585173, 5.96695048575851939400112370573, 6.72787352156314946408273397122, 7.57502462689885957836701094215
