L(s) = 1 | + 0.755·2-s − 1.42·4-s − 5-s + 0.0498·7-s − 2.59·8-s − 0.755·10-s − 4.45·11-s − 2.43·13-s + 0.0376·14-s + 0.898·16-s + 2.48·17-s − 5.16·19-s + 1.42·20-s − 3.36·22-s + 4.99·23-s + 25-s − 1.83·26-s − 0.0711·28-s − 2.59·29-s − 7.31·31-s + 5.86·32-s + 1.87·34-s − 0.0498·35-s + 5.13·37-s − 3.90·38-s + 2.59·40-s − 9.11·41-s + ⋯ |
L(s) = 1 | + 0.534·2-s − 0.714·4-s − 0.447·5-s + 0.0188·7-s − 0.916·8-s − 0.239·10-s − 1.34·11-s − 0.674·13-s + 0.0100·14-s + 0.224·16-s + 0.601·17-s − 1.18·19-s + 0.319·20-s − 0.717·22-s + 1.04·23-s + 0.200·25-s − 0.360·26-s − 0.0134·28-s − 0.481·29-s − 1.31·31-s + 1.03·32-s + 0.321·34-s − 0.00842·35-s + 0.844·37-s − 0.633·38-s + 0.409·40-s − 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.007015393\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.007015393\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 + T \) |
good | 2 | \( 1 - 0.755T + 2T^{2} \) |
| 7 | \( 1 - 0.0498T + 7T^{2} \) |
| 11 | \( 1 + 4.45T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 - 2.48T + 17T^{2} \) |
| 19 | \( 1 + 5.16T + 19T^{2} \) |
| 23 | \( 1 - 4.99T + 23T^{2} \) |
| 29 | \( 1 + 2.59T + 29T^{2} \) |
| 31 | \( 1 + 7.31T + 31T^{2} \) |
| 37 | \( 1 - 5.13T + 37T^{2} \) |
| 41 | \( 1 + 9.11T + 41T^{2} \) |
| 43 | \( 1 + 0.543T + 43T^{2} \) |
| 47 | \( 1 - 9.63T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 2.47T + 61T^{2} \) |
| 67 | \( 1 + 5.36T + 67T^{2} \) |
| 71 | \( 1 + 9.58T + 71T^{2} \) |
| 73 | \( 1 - 2.86T + 73T^{2} \) |
| 79 | \( 1 - 6.62T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 97 | \( 1 - 4.82T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.492116602054792260991867091126, −7.68758182532485681618889774587, −7.10471271710533440491338887079, −5.99458578655158466110516838268, −5.26439824084490635232670197539, −4.78726079315945066614083706944, −3.89701633409337565238951221316, −3.13308294132510319229821892928, −2.21669833011689861232081657837, −0.50813911943900697291309316977,
0.50813911943900697291309316977, 2.21669833011689861232081657837, 3.13308294132510319229821892928, 3.89701633409337565238951221316, 4.78726079315945066614083706944, 5.26439824084490635232670197539, 5.99458578655158466110516838268, 7.10471271710533440491338887079, 7.68758182532485681618889774587, 8.492116602054792260991867091126