| L(s) = 1 | − 0.404·2-s + 3-s − 1.83·4-s − 5-s − 0.404·6-s − 1.25·7-s + 1.55·8-s + 9-s + 0.404·10-s + 6.20·11-s − 1.83·12-s + 1.55·13-s + 0.509·14-s − 15-s + 3.04·16-s + 6.17·17-s − 0.404·18-s + 1.68·19-s + 1.83·20-s − 1.25·21-s − 2.51·22-s + 1.55·24-s + 25-s − 0.630·26-s + 27-s + 2.30·28-s + 3.06·29-s + ⋯ |
| L(s) = 1 | − 0.286·2-s + 0.577·3-s − 0.918·4-s − 0.447·5-s − 0.165·6-s − 0.475·7-s + 0.549·8-s + 0.333·9-s + 0.128·10-s + 1.86·11-s − 0.530·12-s + 0.431·13-s + 0.136·14-s − 0.258·15-s + 0.760·16-s + 1.49·17-s − 0.0954·18-s + 0.385·19-s + 0.410·20-s − 0.274·21-s − 0.535·22-s + 0.317·24-s + 0.200·25-s − 0.123·26-s + 0.192·27-s + 0.436·28-s + 0.568·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.082828722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.082828722\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 0.404T + 2T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 - 6.20T + 11T^{2} \) |
| 13 | \( 1 - 1.55T + 13T^{2} \) |
| 17 | \( 1 - 6.17T + 17T^{2} \) |
| 19 | \( 1 - 1.68T + 19T^{2} \) |
| 29 | \( 1 - 3.06T + 29T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 1.76T + 41T^{2} \) |
| 43 | \( 1 - 1.68T + 43T^{2} \) |
| 47 | \( 1 + 0.384T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 0.978T + 67T^{2} \) |
| 71 | \( 1 + 2.85T + 71T^{2} \) |
| 73 | \( 1 - 4.48T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 - 2.59T + 97T^{2} \) |
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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
−8.011738321880265645787090791554, −7.35401511575064509853125761641, −6.46650661224982470389395400990, −5.89884508362549894777752606114, −4.80365451563882849066520923173, −4.15075987016465149210798469593, −3.57327892089640472055135555956, −2.95497077664243711191741959933, −1.40167254947278623708502880866, −0.855854917771675657316094069247,
0.855854917771675657316094069247, 1.40167254947278623708502880866, 2.95497077664243711191741959933, 3.57327892089640472055135555956, 4.15075987016465149210798469593, 4.80365451563882849066520923173, 5.89884508362549894777752606114, 6.46650661224982470389395400990, 7.35401511575064509853125761641, 8.011738321880265645787090791554
