| L(s) = 1 | − 0.374·2-s − 3-s − 1.85·4-s + 1.32·5-s + 0.374·6-s − 7-s + 1.44·8-s + 9-s − 0.497·10-s + 5.82·11-s + 1.85·12-s + 0.374·14-s − 1.32·15-s + 3.17·16-s − 0.442·17-s − 0.374·18-s + 1.34·19-s − 2.46·20-s + 21-s − 2.18·22-s − 2.67·23-s − 1.44·24-s − 3.23·25-s − 27-s + 1.85·28-s + 3.35·29-s + 0.497·30-s + ⋯ |
| L(s) = 1 | − 0.265·2-s − 0.577·3-s − 0.929·4-s + 0.593·5-s + 0.153·6-s − 0.377·7-s + 0.511·8-s + 0.333·9-s − 0.157·10-s + 1.75·11-s + 0.536·12-s + 0.100·14-s − 0.342·15-s + 0.794·16-s − 0.107·17-s − 0.0883·18-s + 0.308·19-s − 0.551·20-s + 0.218·21-s − 0.465·22-s − 0.558·23-s − 0.295·24-s − 0.647·25-s − 0.192·27-s + 0.351·28-s + 0.622·29-s + 0.0908·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.254181915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.254181915\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.374T + 2T^{2} \) |
| 5 | \( 1 - 1.32T + 5T^{2} \) |
| 11 | \( 1 - 5.82T + 11T^{2} \) |
| 17 | \( 1 + 0.442T + 17T^{2} \) |
| 19 | \( 1 - 1.34T + 19T^{2} \) |
| 23 | \( 1 + 2.67T + 23T^{2} \) |
| 29 | \( 1 - 3.35T + 29T^{2} \) |
| 31 | \( 1 - 8.32T + 31T^{2} \) |
| 37 | \( 1 - 7.93T + 37T^{2} \) |
| 41 | \( 1 + 0.816T + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 - 4.29T + 47T^{2} \) |
| 53 | \( 1 + 1.85T + 53T^{2} \) |
| 59 | \( 1 + 6.16T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 1.64T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 7.20T + 73T^{2} \) |
| 79 | \( 1 + 2.54T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 0.249T + 89T^{2} \) |
| 97 | \( 1 + 3.75T + 97T^{2} \) |
| 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
−8.747676520286840728176733087522, −7.908145348994592486942689705429, −6.99614335106171504355424078037, −6.10836997515833583635779417407, −5.83804224139274646538345047351, −4.52882064205434899349812015403, −4.23155324313815129778177037242, −3.10456887296203496877972712263, −1.65366928945394174546471427719, −0.76353995893802135008619710114,
0.76353995893802135008619710114, 1.65366928945394174546471427719, 3.10456887296203496877972712263, 4.23155324313815129778177037242, 4.52882064205434899349812015403, 5.83804224139274646538345047351, 6.10836997515833583635779417407, 6.99614335106171504355424078037, 7.908145348994592486942689705429, 8.747676520286840728176733087522
