| L(s) = 1 | − 2-s − 3-s + 4-s − 3.56·5-s + 6-s + 3.56·7-s − 8-s + 9-s + 3.56·10-s − 2·11-s − 12-s − 5.12·13-s − 3.56·14-s + 3.56·15-s + 16-s − 2.43·17-s − 18-s − 1.56·19-s − 3.56·20-s − 3.56·21-s + 2·22-s − 23-s + 24-s + 7.68·25-s + 5.12·26-s − 27-s + 3.56·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.59·5-s + 0.408·6-s + 1.34·7-s − 0.353·8-s + 0.333·9-s + 1.12·10-s − 0.603·11-s − 0.288·12-s − 1.42·13-s − 0.951·14-s + 0.919·15-s + 0.250·16-s − 0.591·17-s − 0.235·18-s − 0.358·19-s − 0.796·20-s − 0.777·21-s + 0.426·22-s − 0.208·23-s + 0.204·24-s + 1.53·25-s + 1.00·26-s − 0.192·27-s + 0.673·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4228350211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4228350211\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 + 2.43T + 17T^{2} \) |
| 19 | \( 1 + 1.56T + 19T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 1.56T + 37T^{2} \) |
| 41 | \( 1 + 3.56T + 41T^{2} \) |
| 43 | \( 1 + 0.684T + 43T^{2} \) |
| 47 | \( 1 - 2.43T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 1.56T + 59T^{2} \) |
| 61 | \( 1 + 3.12T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 0.246T + 79T^{2} \) |
| 83 | \( 1 + 2.87T + 83T^{2} \) |
| 89 | \( 1 + 3.12T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.303341589837278045060842141891, −7.65792385984152722402758558428, −7.37105242710467648765308728155, −6.49462963917441556074533770339, −5.25524844064432678307146103636, −4.73343355001403294946793321509, −4.04359928656030092853794585067, −2.80035019239922803942416440636, −1.77875576855483576732294434416, −0.41768954893914248499916994782,
0.41768954893914248499916994782, 1.77875576855483576732294434416, 2.80035019239922803942416440636, 4.04359928656030092853794585067, 4.73343355001403294946793321509, 5.25524844064432678307146103636, 6.49462963917441556074533770339, 7.37105242710467648765308728155, 7.65792385984152722402758558428, 8.303341589837278045060842141891
