| L(s) = 1 | − 2.58·2-s + 4.67·4-s − 2.11·7-s − 6.91·8-s + 4.70·11-s − 3.53·13-s + 5.45·14-s + 8.52·16-s − 6.45·17-s + 0.766·19-s − 12.1·22-s + 5.38·23-s + 9.13·26-s − 9.87·28-s − 8.03·29-s − 31-s − 8.18·32-s + 16.6·34-s + 11.5·37-s − 1.98·38-s + 6.47·41-s − 5.80·43-s + 22.0·44-s − 13.9·46-s + 8.68·47-s − 2.53·49-s − 16.5·52-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.33·4-s − 0.798·7-s − 2.44·8-s + 1.41·11-s − 0.980·13-s + 1.45·14-s + 2.13·16-s − 1.56·17-s + 0.175·19-s − 2.59·22-s + 1.12·23-s + 1.79·26-s − 1.86·28-s − 1.49·29-s − 0.179·31-s − 1.44·32-s + 2.86·34-s + 1.90·37-s − 0.321·38-s + 1.01·41-s − 0.884·43-s + 3.31·44-s − 2.05·46-s + 1.26·47-s − 0.362·49-s − 2.29·52-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 | \( 1 + 2.58T + 2T^{2} \) |
| 7 | \( 1 + 2.11T + 7T^{2} \) |
| 11 | \( 1 - 4.70T + 11T^{2} \) |
| 13 | \( 1 + 3.53T + 13T^{2} \) |
| 17 | \( 1 + 6.45T + 17T^{2} \) |
| 19 | \( 1 - 0.766T + 19T^{2} \) |
| 23 | \( 1 - 5.38T + 23T^{2} \) |
| 29 | \( 1 + 8.03T + 29T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 5.80T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 + 4.05T + 53T^{2} \) |
| 59 | \( 1 + 9.36T + 59T^{2} \) |
| 61 | \( 1 - 3.40T + 61T^{2} \) |
| 67 | \( 1 - 7.82T + 67T^{2} \) |
| 71 | \( 1 - 3.15T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 9.79T + 83T^{2} \) |
| 89 | \( 1 - 0.211T + 89T^{2} \) |
| 97 | \( 1 - 7.13T + 97T^{2} \) |
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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
−7.54645462119620689374222614423, −7.12526633882455568340615554919, −6.50279439223604417774705690345, −5.99461706629741153724448200616, −4.72063669172407225704834866495, −3.76173385184001641899563583326, −2.74564982638254425846652217381, −2.05551949974936532459819002477, −1.01548576201906086182788914910, 0,
1.01548576201906086182788914910, 2.05551949974936532459819002477, 2.74564982638254425846652217381, 3.76173385184001641899563583326, 4.72063669172407225704834866495, 5.99461706629741153724448200616, 6.50279439223604417774705690345, 7.12526633882455568340615554919, 7.54645462119620689374222614423
