| L(s) = 1 | − 2.38·2-s + 3.70·4-s − 1.81·7-s − 4.08·8-s + 0.424·11-s − 1.94·13-s + 4.33·14-s + 2.33·16-s − 2.94·17-s − 5.97·19-s − 1.01·22-s + 2.28·23-s + 4.65·26-s − 6.72·28-s + 4.44·29-s − 31-s + 2.57·32-s + 7.03·34-s − 4.85·37-s + 14.2·38-s + 8.11·41-s + 9.93·43-s + 1.57·44-s − 5.45·46-s + 7.11·47-s − 3.70·49-s − 7.23·52-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 1.85·4-s − 0.685·7-s − 1.44·8-s + 0.128·11-s − 0.540·13-s + 1.15·14-s + 0.584·16-s − 0.714·17-s − 1.37·19-s − 0.216·22-s + 0.476·23-s + 0.913·26-s − 1.27·28-s + 0.825·29-s − 0.179·31-s + 0.455·32-s + 1.20·34-s − 0.798·37-s + 2.31·38-s + 1.26·41-s + 1.51·43-s + 0.237·44-s − 0.804·46-s + 1.03·47-s − 0.529·49-s − 1.00·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 7 | \( 1 + 1.81T + 7T^{2} \) |
| 11 | \( 1 - 0.424T + 11T^{2} \) |
| 13 | \( 1 + 1.94T + 13T^{2} \) |
| 17 | \( 1 + 2.94T + 17T^{2} \) |
| 19 | \( 1 + 5.97T + 19T^{2} \) |
| 23 | \( 1 - 2.28T + 23T^{2} \) |
| 29 | \( 1 - 4.44T + 29T^{2} \) |
| 37 | \( 1 + 4.85T + 37T^{2} \) |
| 41 | \( 1 - 8.11T + 41T^{2} \) |
| 43 | \( 1 - 9.93T + 43T^{2} \) |
| 47 | \( 1 - 7.11T + 47T^{2} \) |
| 53 | \( 1 - 3.16T + 53T^{2} \) |
| 59 | \( 1 - 1.05T + 59T^{2} \) |
| 61 | \( 1 - 6.75T + 61T^{2} \) |
| 67 | \( 1 + 3.33T + 67T^{2} \) |
| 71 | \( 1 + 0.768T + 71T^{2} \) |
| 73 | \( 1 - 9.02T + 73T^{2} \) |
| 79 | \( 1 - 5.54T + 79T^{2} \) |
| 83 | \( 1 + 0.355T + 83T^{2} \) |
| 89 | \( 1 - 9.93T + 89T^{2} \) |
| 97 | \( 1 + 0.875T + 97T^{2} \) |
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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
−7.74167911343013991532390918289, −6.98392147567389320277298767447, −6.58953907386671850974781774069, −5.84707067223574789942706480079, −4.70747980826166933978955788576, −3.87523452166767265149511654099, −2.64942555208972414878569248163, −2.19250330551362978237834798238, −0.962115417452190027500728960620, 0,
0.962115417452190027500728960620, 2.19250330551362978237834798238, 2.64942555208972414878569248163, 3.87523452166767265149511654099, 4.70747980826166933978955788576, 5.84707067223574789942706480079, 6.58953907386671850974781774069, 6.98392147567389320277298767447, 7.74167911343013991532390918289
