L(s) = 1 | + 8.26·2-s + 12.5·3-s + 36.2·4-s + 25·5-s + 104.·6-s + 63.2·7-s + 35.1·8-s − 84.3·9-s + 206.·10-s + 468.·11-s + 456.·12-s + 1.12e3·13-s + 522.·14-s + 314.·15-s − 869.·16-s − 601.·17-s − 696.·18-s − 608.·19-s + 906.·20-s + 796.·21-s + 3.87e3·22-s − 529·23-s + 442.·24-s + 625·25-s + 9.27e3·26-s − 4.12e3·27-s + 2.29e3·28-s + ⋯ |
L(s) = 1 | + 1.46·2-s + 0.807·3-s + 1.13·4-s + 0.447·5-s + 1.17·6-s + 0.487·7-s + 0.193·8-s − 0.347·9-s + 0.653·10-s + 1.16·11-s + 0.915·12-s + 1.84·13-s + 0.712·14-s + 0.361·15-s − 0.849·16-s − 0.504·17-s − 0.507·18-s − 0.386·19-s + 0.506·20-s + 0.394·21-s + 1.70·22-s − 0.208·23-s + 0.156·24-s + 0.200·25-s + 2.69·26-s − 1.08·27-s + 0.552·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(5.798454209\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.798454209\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 25T \) |
| 23 | \( 1 + 529T \) |
good | 2 | \( 1 - 8.26T + 32T^{2} \) |
| 3 | \( 1 - 12.5T + 243T^{2} \) |
| 7 | \( 1 - 63.2T + 1.68e4T^{2} \) |
| 11 | \( 1 - 468.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.12e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 601.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 608.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 1.89e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 134.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.54e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.46e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.43e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.33e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.76e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.21e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.90e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.56e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.78e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.00e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.00e4T + 8.58e9T^{2} \) |
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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
−13.06209095553742599980563855271, −11.76111481515960880635121470523, −10.96929301776223921314602282501, −9.160183944484929070952685294093, −8.436938589101688658367478028547, −6.61296004860623047775555827151, −5.73158190451318165764207592731, −4.22560003337489821682573459456, −3.28441480810567847629992578349, −1.78226617409113099548997557121,
1.78226617409113099548997557121, 3.28441480810567847629992578349, 4.22560003337489821682573459456, 5.73158190451318165764207592731, 6.61296004860623047775555827151, 8.436938589101688658367478028547, 9.160183944484929070952685294093, 10.96929301776223921314602282501, 11.76111481515960880635121470523, 13.06209095553742599980563855271