L(s) = 1 | − 2-s + 4-s − 2.22·5-s − 0.652·7-s − 8-s + 2.22·10-s + 1.53·11-s + 0.467·13-s + 0.652·14-s + 16-s + 2.10·17-s − 2.22·20-s − 1.53·22-s − 5.90·23-s − 0.0418·25-s − 0.467·26-s − 0.652·28-s − 8.33·29-s + 8.63·31-s − 32-s − 2.10·34-s + 1.45·35-s + 4.67·37-s + 2.22·40-s + 3.47·41-s + 10.2·43-s + 1.53·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.995·5-s − 0.246·7-s − 0.353·8-s + 0.704·10-s + 0.461·11-s + 0.129·13-s + 0.174·14-s + 0.250·16-s + 0.510·17-s − 0.497·20-s − 0.326·22-s − 1.23·23-s − 0.00837·25-s − 0.0917·26-s − 0.123·28-s − 1.54·29-s + 1.55·31-s − 0.176·32-s − 0.361·34-s + 0.245·35-s + 0.768·37-s + 0.352·40-s + 0.542·41-s + 1.56·43-s + 0.230·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 2.22T + 5T^{2} \) |
| 7 | \( 1 + 0.652T + 7T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 - 0.467T + 13T^{2} \) |
| 17 | \( 1 - 2.10T + 17T^{2} \) |
| 23 | \( 1 + 5.90T + 23T^{2} \) |
| 29 | \( 1 + 8.33T + 29T^{2} \) |
| 31 | \( 1 - 8.63T + 31T^{2} \) |
| 37 | \( 1 - 4.67T + 37T^{2} \) |
| 41 | \( 1 - 3.47T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 4.63T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 + 0.248T + 67T^{2} \) |
| 71 | \( 1 + 4.44T + 71T^{2} \) |
| 73 | \( 1 + 9.09T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 3.70T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 2.46T + 97T^{2} \) |
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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
−7.65574445211818549620194626210, −7.31133136156651192089727612828, −6.19183367954046071205937598891, −5.89022044829534298197370366106, −4.58490468497241673907751222306, −3.94037631869557974020537035795, −3.21242043133111311202655672689, −2.20658112104966058744225493887, −1.08508330011487458343229041718, 0,
1.08508330011487458343229041718, 2.20658112104966058744225493887, 3.21242043133111311202655672689, 3.94037631869557974020537035795, 4.58490468497241673907751222306, 5.89022044829534298197370366106, 6.19183367954046071205937598891, 7.31133136156651192089727612828, 7.65574445211818549620194626210