| L(s) = 1 | − 0.528·3-s − 5·5-s − 17.5·7-s − 26.7·9-s − 24.1·11-s − 79.1·13-s + 2.64·15-s − 6.40·17-s − 127.·19-s + 9.26·21-s − 52.0·23-s + 25·25-s + 28.3·27-s + 29·29-s + 89.9·31-s + 12.7·33-s + 87.6·35-s − 71.2·37-s + 41.8·39-s + 34.0·41-s + 355.·43-s + 133.·45-s − 600.·47-s − 35.7·49-s + 3.38·51-s + 738.·53-s + 120.·55-s + ⋯ |
| L(s) = 1 | − 0.101·3-s − 0.447·5-s − 0.946·7-s − 0.989·9-s − 0.662·11-s − 1.68·13-s + 0.0454·15-s − 0.0913·17-s − 1.54·19-s + 0.0962·21-s − 0.472·23-s + 0.200·25-s + 0.202·27-s + 0.185·29-s + 0.521·31-s + 0.0673·33-s + 0.423·35-s − 0.316·37-s + 0.171·39-s + 0.129·41-s + 1.26·43-s + 0.442·45-s − 1.86·47-s − 0.104·49-s + 0.00929·51-s + 1.91·53-s + 0.296·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2347252654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2347252654\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 0.528T + 27T^{2} \) |
| 7 | \( 1 + 17.5T + 343T^{2} \) |
| 11 | \( 1 + 24.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 79.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.40T + 4.91e3T^{2} \) |
| 19 | \( 1 + 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 52.0T + 1.21e4T^{2} \) |
| 31 | \( 1 - 89.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 71.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 34.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 355.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 600.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 738.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 151.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 581.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 74.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 281.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 578.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 549.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 76.0T + 5.71e5T^{2} \) |
| 89 | \( 1 + 406.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.16e3T + 9.12e5T^{2} \) |
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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
−9.463269640967706047176590201820, −8.541211019965665049133479188100, −7.81993325378592267584047433273, −6.88293269621421063884655919591, −6.10639413846455198459391047531, −5.13695553550460893813683844552, −4.22153237544055161836660713019, −2.99829569636976870056863295574, −2.32630673924041099380254854433, −0.22989929427295113686550148219,
0.22989929427295113686550148219, 2.32630673924041099380254854433, 2.99829569636976870056863295574, 4.22153237544055161836660713019, 5.13695553550460893813683844552, 6.10639413846455198459391047531, 6.88293269621421063884655919591, 7.81993325378592267584047433273, 8.541211019965665049133479188100, 9.463269640967706047176590201820
