| L(s) = 1 | − 0.458·2-s − 1.79·4-s + 0.388·5-s − 4.22·7-s + 1.73·8-s − 0.177·10-s − 2.44·11-s + 0.942·13-s + 1.93·14-s + 2.78·16-s + 2.35·17-s + 2.89·19-s − 0.695·20-s + 1.11·22-s − 23-s − 4.84·25-s − 0.431·26-s + 7.56·28-s − 29-s + 7.75·31-s − 4.74·32-s − 1.07·34-s − 1.63·35-s − 2.82·37-s − 1.32·38-s + 0.674·40-s − 5.25·41-s + ⋯ |
| L(s) = 1 | − 0.324·2-s − 0.895·4-s + 0.173·5-s − 1.59·7-s + 0.614·8-s − 0.0562·10-s − 0.736·11-s + 0.261·13-s + 0.517·14-s + 0.696·16-s + 0.571·17-s + 0.663·19-s − 0.155·20-s + 0.238·22-s − 0.208·23-s − 0.969·25-s − 0.0847·26-s + 1.42·28-s − 0.185·29-s + 1.39·31-s − 0.839·32-s − 0.185·34-s − 0.277·35-s − 0.464·37-s − 0.215·38-s + 0.106·40-s − 0.821·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5821096740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5821096740\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.458T + 2T^{2} \) |
| 5 | \( 1 - 0.388T + 5T^{2} \) |
| 7 | \( 1 + 4.22T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 0.942T + 13T^{2} \) |
| 17 | \( 1 - 2.35T + 17T^{2} \) |
| 19 | \( 1 - 2.89T + 19T^{2} \) |
| 31 | \( 1 - 7.75T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 + 5.25T + 41T^{2} \) |
| 43 | \( 1 + 9.79T + 43T^{2} \) |
| 47 | \( 1 + 9.98T + 47T^{2} \) |
| 53 | \( 1 + 6.00T + 53T^{2} \) |
| 59 | \( 1 + 8.95T + 59T^{2} \) |
| 61 | \( 1 - 8.75T + 61T^{2} \) |
| 67 | \( 1 + 8.58T + 67T^{2} \) |
| 71 | \( 1 + 6.86T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 7.05T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 8.73T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−8.156053999075191479492636532334, −7.50663992870350991100849419188, −6.62487246577722882704456953659, −5.95577527517137320286271956026, −5.25691720802474590623545379268, −4.46786525441096792245189826057, −3.39813767592624359402769651561, −3.11307979255060718309729326151, −1.68740850111438368704854908056, −0.42167530920639288242646385399,
0.42167530920639288242646385399, 1.68740850111438368704854908056, 3.11307979255060718309729326151, 3.39813767592624359402769651561, 4.46786525441096792245189826057, 5.25691720802474590623545379268, 5.95577527517137320286271956026, 6.62487246577722882704456953659, 7.50663992870350991100849419188, 8.156053999075191479492636532334
