L(s) = 1 | − 0.910·2-s − 3-s − 1.17·4-s + 5-s + 0.910·6-s − 2.43·7-s + 2.88·8-s + 9-s − 0.910·10-s − 3.76·11-s + 1.17·12-s − 4.94·13-s + 2.21·14-s − 15-s − 0.289·16-s + 2.79·17-s − 0.910·18-s − 3.08·19-s − 1.17·20-s + 2.43·21-s + 3.42·22-s − 2.88·24-s + 25-s + 4.50·26-s − 27-s + 2.84·28-s + 3.02·29-s + ⋯ |
L(s) = 1 | − 0.644·2-s − 0.577·3-s − 0.585·4-s + 0.447·5-s + 0.371·6-s − 0.918·7-s + 1.02·8-s + 0.333·9-s − 0.288·10-s − 1.13·11-s + 0.337·12-s − 1.37·13-s + 0.591·14-s − 0.258·15-s − 0.0724·16-s + 0.677·17-s − 0.214·18-s − 0.708·19-s − 0.261·20-s + 0.530·21-s + 0.730·22-s − 0.589·24-s + 0.200·25-s + 0.884·26-s − 0.192·27-s + 0.537·28-s + 0.562·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + 0.910T + 2T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 + 3.76T + 11T^{2} \) |
| 13 | \( 1 + 4.94T + 13T^{2} \) |
| 17 | \( 1 - 2.79T + 17T^{2} \) |
| 19 | \( 1 + 3.08T + 19T^{2} \) |
| 29 | \( 1 - 3.02T + 29T^{2} \) |
| 31 | \( 1 - 7.55T + 31T^{2} \) |
| 37 | \( 1 - 0.0646T + 37T^{2} \) |
| 41 | \( 1 + 5.77T + 41T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 + 2.31T + 47T^{2} \) |
| 53 | \( 1 - 3.16T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 6.33T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 1.33T + 73T^{2} \) |
| 79 | \( 1 + 7.35T + 79T^{2} \) |
| 83 | \( 1 - 2.04T + 83T^{2} \) |
| 89 | \( 1 - 7.99T + 89T^{2} \) |
| 97 | \( 1 + 0.335T + 97T^{2} \) |
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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
−7.53133766847274885283046545895, −6.88988857800484175911490489331, −6.13882237503305608391650331554, −5.28889151004961843178283940769, −4.89439829610427156337191531926, −4.03778368215477035457966772723, −2.93585345974428261327480213497, −2.18650935118359513947501540161, −0.857958940786670871451665443102, 0,
0.857958940786670871451665443102, 2.18650935118359513947501540161, 2.93585345974428261327480213497, 4.03778368215477035457966772723, 4.89439829610427156337191531926, 5.28889151004961843178283940769, 6.13882237503305608391650331554, 6.88988857800484175911490489331, 7.53133766847274885283046545895