| L(s) = 1 | − 0.618·2-s + 3-s − 1.61·4-s − 0.618·6-s + 7-s + 2.23·8-s − 2·9-s + 4.47·11-s − 1.61·12-s − 0.236·13-s − 0.618·14-s + 1.85·16-s + 1.23·18-s − 7.23·19-s + 21-s − 2.76·22-s + 23-s + 2.23·24-s + 0.145·26-s − 5·27-s − 1.61·28-s − 1.47·29-s − 9·31-s − 5.61·32-s + 4.47·33-s + 3.23·36-s + 5.70·37-s + ⋯ |
| L(s) = 1 | − 0.437·2-s + 0.577·3-s − 0.809·4-s − 0.252·6-s + 0.377·7-s + 0.790·8-s − 0.666·9-s + 1.34·11-s − 0.467·12-s − 0.0654·13-s − 0.165·14-s + 0.463·16-s + 0.291·18-s − 1.66·19-s + 0.218·21-s − 0.589·22-s + 0.208·23-s + 0.456·24-s + 0.0286·26-s − 0.962·27-s − 0.305·28-s − 0.273·29-s − 1.61·31-s − 0.993·32-s + 0.778·33-s + 0.539·36-s + 0.938·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 + 0.236T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 + 9T + 31T^{2} \) |
| 37 | \( 1 - 5.70T + 37T^{2} \) |
| 41 | \( 1 + 2.23T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 - 3.47T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 6.70T + 73T^{2} \) |
| 79 | \( 1 + 7.23T + 79T^{2} \) |
| 83 | \( 1 + 6.47T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 + 3.70T + 97T^{2} \) |
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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
−8.350329933734952409859327177506, −7.59722754166431819530618122077, −6.70549368654592222642490328159, −5.86541483197722003307724598324, −4.98968771633073778616943756383, −4.08238891614911344155956974363, −3.61938829907485201539472834940, −2.32208641849278377881195589094, −1.39833687032560782469157297142, 0,
1.39833687032560782469157297142, 2.32208641849278377881195589094, 3.61938829907485201539472834940, 4.08238891614911344155956974363, 4.98968771633073778616943756383, 5.86541483197722003307724598324, 6.70549368654592222642490328159, 7.59722754166431819530618122077, 8.350329933734952409859327177506
