| L(s) = 1 | + 1.61·2-s + 3-s + 0.618·4-s + 1.61·6-s − 4.23·7-s − 2.23·8-s + 9-s − 5.23·11-s + 0.618·12-s + 1.23·13-s − 6.85·14-s − 4.85·16-s + 2.47·17-s + 1.61·18-s + 19-s − 4.23·21-s − 8.47·22-s − 0.472·23-s − 2.23·24-s + 2.00·26-s + 27-s − 2.61·28-s − 0.527·29-s − 8·31-s − 3.38·32-s − 5.23·33-s + 4.00·34-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.577·3-s + 0.309·4-s + 0.660·6-s − 1.60·7-s − 0.790·8-s + 0.333·9-s − 1.57·11-s + 0.178·12-s + 0.342·13-s − 1.83·14-s − 1.21·16-s + 0.599·17-s + 0.381·18-s + 0.229·19-s − 0.924·21-s − 1.80·22-s − 0.0984·23-s − 0.456·24-s + 0.392·26-s + 0.192·27-s − 0.494·28-s − 0.0980·29-s − 1.43·31-s − 0.597·32-s − 0.911·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 23 | \( 1 + 0.472T + 23T^{2} \) |
| 29 | \( 1 + 0.527T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 9.23T + 37T^{2} \) |
| 41 | \( 1 + 7.47T + 41T^{2} \) |
| 43 | \( 1 + 4.94T + 43T^{2} \) |
| 47 | \( 1 - 0.763T + 47T^{2} \) |
| 53 | \( 1 + T + 53T^{2} \) |
| 59 | \( 1 - 6.70T + 59T^{2} \) |
| 61 | \( 1 - 1.47T + 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 5.70T + 83T^{2} \) |
| 89 | \( 1 + 5T + 89T^{2} \) |
| 97 | \( 1 - 9.70T + 97T^{2} \) |
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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
−9.173304439313308628419290032566, −8.384583687218046965842645639274, −7.33813706161780314131634873777, −6.55001549925351331738774475117, −5.60716530995683553534752010767, −5.02413268808905679111821291713, −3.57756546733495541605894688168, −3.37605293531069060471817792327, −2.33127289349717856914364225476, 0,
2.33127289349717856914364225476, 3.37605293531069060471817792327, 3.57756546733495541605894688168, 5.02413268808905679111821291713, 5.60716530995683553534752010767, 6.55001549925351331738774475117, 7.33813706161780314131634873777, 8.384583687218046965842645639274, 9.173304439313308628419290032566
