| L(s) = 1 | − 2.44·2-s + 3-s + 3.99·4-s + 5-s − 2.44·6-s + 1.75·7-s − 4.87·8-s + 9-s − 2.44·10-s + 1.55·11-s + 3.99·12-s − 5.20·13-s − 4.30·14-s + 15-s + 3.95·16-s − 2.47·17-s − 2.44·18-s − 1.85·19-s + 3.99·20-s + 1.75·21-s − 3.79·22-s − 4.87·24-s + 25-s + 12.7·26-s + 27-s + 7.01·28-s − 10.2·29-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 0.577·3-s + 1.99·4-s + 0.447·5-s − 0.999·6-s + 0.664·7-s − 1.72·8-s + 0.333·9-s − 0.774·10-s + 0.467·11-s + 1.15·12-s − 1.44·13-s − 1.14·14-s + 0.258·15-s + 0.987·16-s − 0.599·17-s − 0.576·18-s − 0.424·19-s + 0.892·20-s + 0.383·21-s − 0.809·22-s − 0.995·24-s + 0.200·25-s + 2.49·26-s + 0.192·27-s + 1.32·28-s − 1.91·29-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 7 | \( 1 - 1.75T + 7T^{2} \) |
| 11 | \( 1 - 1.55T + 11T^{2} \) |
| 13 | \( 1 + 5.20T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 2.30T + 31T^{2} \) |
| 37 | \( 1 - 6.56T + 37T^{2} \) |
| 41 | \( 1 - 3.29T + 41T^{2} \) |
| 43 | \( 1 + 4.74T + 43T^{2} \) |
| 47 | \( 1 - 7.93T + 47T^{2} \) |
| 53 | \( 1 + 8.51T + 53T^{2} \) |
| 59 | \( 1 - 0.268T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 8.79T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 - 6.52T + 79T^{2} \) |
| 83 | \( 1 + 2.69T + 83T^{2} \) |
| 89 | \( 1 - 3.86T + 89T^{2} \) |
| 97 | \( 1 - 9.26T + 97T^{2} \) |
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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
−7.85768440940384627803683330082, −7.03394887631853737262423122759, −6.58076107866088772232069103901, −5.54116535435554898341090570874, −4.68992955240927034016134347241, −3.74765492645953129586195606215, −2.43241776605630186931379920211, −2.16068426431266249206491514361, −1.26047203888461618099097686437, 0,
1.26047203888461618099097686437, 2.16068426431266249206491514361, 2.43241776605630186931379920211, 3.74765492645953129586195606215, 4.68992955240927034016134347241, 5.54116535435554898341090570874, 6.58076107866088772232069103901, 7.03394887631853737262423122759, 7.85768440940384627803683330082
