L(s) = 1 | − 0.698·2-s + 3-s − 1.51·4-s + 5-s − 0.698·6-s + 4.99·7-s + 2.45·8-s + 9-s − 0.698·10-s − 1.56·11-s − 1.51·12-s − 13-s − 3.48·14-s + 15-s + 1.31·16-s + 0.300·17-s − 0.698·18-s − 6.71·19-s − 1.51·20-s + 4.99·21-s + 1.09·22-s + 3.87·23-s + 2.45·24-s + 25-s + 0.698·26-s + 27-s − 7.55·28-s + ⋯ |
L(s) = 1 | − 0.493·2-s + 0.577·3-s − 0.756·4-s + 0.447·5-s − 0.285·6-s + 1.88·7-s + 0.867·8-s + 0.333·9-s − 0.220·10-s − 0.473·11-s − 0.436·12-s − 0.277·13-s − 0.932·14-s + 0.258·15-s + 0.328·16-s + 0.0728·17-s − 0.164·18-s − 1.54·19-s − 0.338·20-s + 1.09·21-s + 0.233·22-s + 0.807·23-s + 0.500·24-s + 0.200·25-s + 0.136·26-s + 0.192·27-s − 1.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.140789559\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.140789559\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + 0.698T + 2T^{2} \) |
| 7 | \( 1 - 4.99T + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 17 | \( 1 - 0.300T + 17T^{2} \) |
| 19 | \( 1 + 6.71T + 19T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 + 3.73T + 29T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 + 2.60T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 5.68T + 47T^{2} \) |
| 53 | \( 1 - 6.83T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 8.54T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 3.14T + 73T^{2} \) |
| 79 | \( 1 - 6.81T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 - 4.84T + 89T^{2} \) |
| 97 | \( 1 + 7.32T + 97T^{2} \) |
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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
−8.192630924861699460911092602596, −7.61032088809800906412114007451, −7.01492764812480644067262555641, −5.70120882061976864873673196112, −5.11151599442579431327673975502, −4.47639830333820609253035422168, −3.84072587833722278446981407206, −2.42196752815811495114885836617, −1.85703480034152668473404898923, −0.846087140066560604539552811700,
0.846087140066560604539552811700, 1.85703480034152668473404898923, 2.42196752815811495114885836617, 3.84072587833722278446981407206, 4.47639830333820609253035422168, 5.11151599442579431327673975502, 5.70120882061976864873673196112, 7.01492764812480644067262555641, 7.61032088809800906412114007451, 8.192630924861699460911092602596