L(s) = 1 | − 0.214·2-s − 3.36·3-s − 1.95·4-s − 5-s + 0.723·6-s − 7-s + 0.849·8-s + 8.34·9-s + 0.214·10-s − 5.90·11-s + 6.58·12-s − 2.24·13-s + 0.214·14-s + 3.36·15-s + 3.72·16-s + 7.21·17-s − 1.79·18-s + 0.212·19-s + 1.95·20-s + 3.36·21-s + 1.26·22-s + 5.27·23-s − 2.86·24-s + 25-s + 0.482·26-s − 18.0·27-s + 1.95·28-s + ⋯ |
L(s) = 1 | − 0.151·2-s − 1.94·3-s − 0.976·4-s − 0.447·5-s + 0.295·6-s − 0.377·7-s + 0.300·8-s + 2.78·9-s + 0.0679·10-s − 1.78·11-s + 1.90·12-s − 0.623·13-s + 0.0573·14-s + 0.869·15-s + 0.931·16-s + 1.74·17-s − 0.422·18-s + 0.0487·19-s + 0.436·20-s + 0.735·21-s + 0.270·22-s + 1.09·23-s − 0.583·24-s + 0.200·25-s + 0.0947·26-s − 3.46·27-s + 0.369·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4097202336\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4097202336\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 + 0.214T + 2T^{2} \) |
| 3 | \( 1 + 3.36T + 3T^{2} \) |
| 11 | \( 1 + 5.90T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 - 7.21T + 17T^{2} \) |
| 19 | \( 1 - 0.212T + 19T^{2} \) |
| 23 | \( 1 - 5.27T + 23T^{2} \) |
| 29 | \( 1 + 2.08T + 29T^{2} \) |
| 31 | \( 1 - 5.05T + 31T^{2} \) |
| 37 | \( 1 - 8.04T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 0.0414T + 47T^{2} \) |
| 53 | \( 1 - 5.38T + 53T^{2} \) |
| 59 | \( 1 + 9.03T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 3.53T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 6.55T + 73T^{2} \) |
| 79 | \( 1 + 8.07T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 3.46T + 97T^{2} \) |
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
−7.58049614588900168849595229428, −7.33299801469785632594011604133, −6.23116611648786708945945282809, −5.56277432111451599297008260585, −5.10599590046550575421412486788, −4.65866249181101549650191674397, −3.76686173960876007073148942583, −2.75911004066445493921313233045, −1.11109023750489093590253512405, −0.45006833863326383909099398557,
0.45006833863326383909099398557, 1.11109023750489093590253512405, 2.75911004066445493921313233045, 3.76686173960876007073148942583, 4.65866249181101549650191674397, 5.10599590046550575421412486788, 5.56277432111451599297008260585, 6.23116611648786708945945282809, 7.33299801469785632594011604133, 7.58049614588900168849595229428