| L(s) = 1 | − 1.87·2-s + 3-s + 1.53·4-s + 0.879·5-s − 1.87·6-s − 2.87·7-s + 0.879·8-s + 9-s − 1.65·10-s − 1.83·11-s + 1.53·12-s + 2.75·13-s + 5.41·14-s + 0.879·15-s − 4.71·16-s − 7.10·17-s − 1.87·18-s + 1.34·20-s − 2.87·21-s + 3.45·22-s + 6.59·23-s + 0.879·24-s − 4.22·25-s − 5.18·26-s + 27-s − 4.41·28-s + 3.12·29-s + ⋯ |
| L(s) = 1 | − 1.32·2-s + 0.577·3-s + 0.766·4-s + 0.393·5-s − 0.767·6-s − 1.08·7-s + 0.310·8-s + 0.333·9-s − 0.522·10-s − 0.554·11-s + 0.442·12-s + 0.765·13-s + 1.44·14-s + 0.227·15-s − 1.17·16-s − 1.72·17-s − 0.442·18-s + 0.301·20-s − 0.628·21-s + 0.736·22-s + 1.37·23-s + 0.179·24-s − 0.845·25-s − 1.01·26-s + 0.192·27-s − 0.833·28-s + 0.580·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.87T + 2T^{2} \) |
| 5 | \( 1 - 0.879T + 5T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 + 1.83T + 11T^{2} \) |
| 13 | \( 1 - 2.75T + 13T^{2} \) |
| 17 | \( 1 + 7.10T + 17T^{2} \) |
| 23 | \( 1 - 6.59T + 23T^{2} \) |
| 29 | \( 1 - 3.12T + 29T^{2} \) |
| 31 | \( 1 + 7.65T + 31T^{2} \) |
| 37 | \( 1 + 2.83T + 37T^{2} \) |
| 41 | \( 1 + 3.98T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 2.20T + 47T^{2} \) |
| 53 | \( 1 + 2.70T + 53T^{2} \) |
| 59 | \( 1 - 8.41T + 59T^{2} \) |
| 61 | \( 1 - 0.615T + 61T^{2} \) |
| 67 | \( 1 - 3.67T + 67T^{2} \) |
| 71 | \( 1 - 7.45T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 1.61T + 79T^{2} \) |
| 83 | \( 1 - 0.985T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 5.90T + 97T^{2} \) |
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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
−9.323809592767699733041422884728, −8.804554995447597577420281045048, −8.140729276835605694161798542535, −6.98265403513182835066762774541, −6.59289796508453320225512635831, −5.21405868734692212836788368330, −3.92973365701156412986090157253, −2.76558405894223687654421783072, −1.67527569965281031494178029870, 0,
1.67527569965281031494178029870, 2.76558405894223687654421783072, 3.92973365701156412986090157253, 5.21405868734692212836788368330, 6.59289796508453320225512635831, 6.98265403513182835066762774541, 8.140729276835605694161798542535, 8.804554995447597577420281045048, 9.323809592767699733041422884728
