| L(s) = 1 | − 2.64·2-s + 3-s + 5.00·4-s + 5-s − 2.64·6-s − 3.64·7-s − 7.93·8-s + 9-s − 2.64·10-s + 5.64·11-s + 5.00·12-s + 5.64·13-s + 9.64·14-s + 15-s + 11.0·16-s − 4·17-s − 2.64·18-s + 5.00·20-s − 3.64·21-s − 14.9·22-s − 1.29·23-s − 7.93·24-s + 25-s − 14.9·26-s + 27-s − 18.2·28-s + 6.93·29-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 0.577·3-s + 2.50·4-s + 0.447·5-s − 1.08·6-s − 1.37·7-s − 2.80·8-s + 0.333·9-s − 0.836·10-s + 1.70·11-s + 1.44·12-s + 1.56·13-s + 2.57·14-s + 0.258·15-s + 2.75·16-s − 0.970·17-s − 0.623·18-s + 1.11·20-s − 0.795·21-s − 3.18·22-s − 0.269·23-s − 1.62·24-s + 0.200·25-s − 2.92·26-s + 0.192·27-s − 3.44·28-s + 1.28·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.137338761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.137338761\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 7 | \( 1 + 3.64T + 7T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 - 5.64T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 23 | \( 1 + 1.29T + 23T^{2} \) |
| 29 | \( 1 - 6.93T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 1.64T + 37T^{2} \) |
| 41 | \( 1 - 4.35T + 41T^{2} \) |
| 43 | \( 1 - 0.354T + 43T^{2} \) |
| 47 | \( 1 - 9.29T + 47T^{2} \) |
| 53 | \( 1 + 0.708T + 53T^{2} \) |
| 59 | \( 1 + 0.708T + 59T^{2} \) |
| 61 | \( 1 + 0.708T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 1.06T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.577154902594789531918917152613, −7.60066817409623747775818506043, −6.71265043129274183556570302753, −6.46841948018535075575916695413, −5.92680962740223158359931565238, −4.09839966178114707604350028136, −3.40874528039129945231204597838, −2.51850215320828028210424600792, −1.59273821173736288635935958014, −0.77688744277918160687460269859,
0.77688744277918160687460269859, 1.59273821173736288635935958014, 2.51850215320828028210424600792, 3.40874528039129945231204597838, 4.09839966178114707604350028136, 5.92680962740223158359931565238, 6.46841948018535075575916695413, 6.71265043129274183556570302753, 7.60066817409623747775818506043, 8.577154902594789531918917152613
