| L(s) = 1 | − 1.29·5-s + 7-s − 2·11-s + 5.71·13-s − 1.29·17-s − 19-s − 2·23-s − 3.31·25-s + 10.7·29-s − 3.71·31-s − 1.29·35-s − 0.599·37-s + 4·41-s + 10.3·43-s − 10.1·47-s + 49-s − 0.700·53-s + 2.59·55-s + 3.40·61-s − 7.42·65-s − 2.59·67-s + 0.700·71-s + 13.4·73-s − 2·77-s − 6.02·79-s + 12.7·83-s + 1.68·85-s + ⋯ |
| L(s) = 1 | − 0.581·5-s + 0.377·7-s − 0.603·11-s + 1.58·13-s − 0.315·17-s − 0.229·19-s − 0.417·23-s − 0.662·25-s + 1.99·29-s − 0.666·31-s − 0.219·35-s − 0.0985·37-s + 0.624·41-s + 1.57·43-s − 1.47·47-s + 0.142·49-s − 0.0961·53-s + 0.350·55-s + 0.435·61-s − 0.920·65-s − 0.317·67-s + 0.0831·71-s + 1.57·73-s − 0.227·77-s − 0.677·79-s + 1.39·83-s + 0.183·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.727220169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.727220169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 1.29T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 5.71T + 13T^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 10.7T + 29T^{2} \) |
| 31 | \( 1 + 3.71T + 31T^{2} \) |
| 37 | \( 1 + 0.599T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 0.700T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 3.40T + 61T^{2} \) |
| 67 | \( 1 + 2.59T + 67T^{2} \) |
| 71 | \( 1 - 0.700T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 6.02T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.139254648520164096402630087330, −7.84207010344993905814159649291, −6.78198260623656034077653943532, −6.15378104366534507591352389367, −5.36870529312430896292562431256, −4.43992733834217977085799158121, −3.85133339867108035484888162138, −2.94396996101143365399959939923, −1.88759697978961040943738250722, −0.73338856144486033953605719948,
0.73338856144486033953605719948, 1.88759697978961040943738250722, 2.94396996101143365399959939923, 3.85133339867108035484888162138, 4.43992733834217977085799158121, 5.36870529312430896292562431256, 6.15378104366534507591352389367, 6.78198260623656034077653943532, 7.84207010344993905814159649291, 8.139254648520164096402630087330
