| L(s) = 1 | − 5-s − 5.24·7-s − 5.14·11-s + 2.10·13-s − 5.14·17-s − 7.24·19-s − 6.24·23-s + 25-s + 4.24·29-s − 3.14·31-s + 5.24·35-s − 1.24·37-s − 2.20·41-s + 6.03·43-s + 10.2·47-s + 20.5·49-s − 4·53-s + 5.14·55-s + 6.49·59-s − 1.24·61-s − 2.10·65-s − 5.03·67-s + 1.79·71-s − 11.7·73-s + 26.9·77-s + 6.10·79-s − 14.7·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.98·7-s − 1.55·11-s + 0.583·13-s − 1.24·17-s − 1.66·19-s − 1.30·23-s + 0.200·25-s + 0.788·29-s − 0.564·31-s + 0.886·35-s − 0.205·37-s − 0.344·41-s + 0.921·43-s + 1.49·47-s + 2.93·49-s − 0.549·53-s + 0.693·55-s + 0.845·59-s − 0.159·61-s − 0.260·65-s − 0.615·67-s + 0.212·71-s − 1.37·73-s + 3.07·77-s + 0.686·79-s − 1.61·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3043724055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3043724055\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 5.24T + 7T^{2} \) |
| 11 | \( 1 + 5.14T + 11T^{2} \) |
| 13 | \( 1 - 2.10T + 13T^{2} \) |
| 17 | \( 1 + 5.14T + 17T^{2} \) |
| 19 | \( 1 + 7.24T + 19T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 + 3.14T + 31T^{2} \) |
| 37 | \( 1 + 1.24T + 37T^{2} \) |
| 41 | \( 1 + 2.20T + 41T^{2} \) |
| 43 | \( 1 - 6.03T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 - 6.49T + 59T^{2} \) |
| 61 | \( 1 + 1.24T + 61T^{2} \) |
| 67 | \( 1 + 5.03T + 67T^{2} \) |
| 71 | \( 1 - 1.79T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 6.10T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 8.28T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.517558936051217141799438545555, −7.60611868672989344487272455390, −6.83133099642967895569491602491, −6.22663733187024959003729787283, −5.64969489314850300841663638416, −4.39731747772460396022716806299, −3.85507768623621205505526021596, −2.85852403481236599797121329819, −2.25542684492266224368528799853, −0.28370810090326189843679171345,
0.28370810090326189843679171345, 2.25542684492266224368528799853, 2.85852403481236599797121329819, 3.85507768623621205505526021596, 4.39731747772460396022716806299, 5.64969489314850300841663638416, 6.22663733187024959003729787283, 6.83133099642967895569491602491, 7.60611868672989344487272455390, 8.517558936051217141799438545555
