| L(s) = 1 | − 0.658·3-s − 2.39·5-s + 4.45·7-s − 2.56·9-s + 11-s + 3.38·13-s + 1.57·15-s − 5.17·17-s + 19-s − 2.93·21-s + 0.851·23-s + 0.750·25-s + 3.66·27-s + 7.62·29-s + 6.34·31-s − 0.658·33-s − 10.6·35-s + 9.00·37-s − 2.22·39-s − 3.15·41-s + 5.24·43-s + 6.15·45-s + 8.77·47-s + 12.8·49-s + 3.40·51-s + 9.81·53-s − 2.39·55-s + ⋯ |
| L(s) = 1 | − 0.380·3-s − 1.07·5-s + 1.68·7-s − 0.855·9-s + 0.301·11-s + 0.938·13-s + 0.407·15-s − 1.25·17-s + 0.229·19-s − 0.640·21-s + 0.177·23-s + 0.150·25-s + 0.705·27-s + 1.41·29-s + 1.14·31-s − 0.114·33-s − 1.80·35-s + 1.48·37-s − 0.356·39-s − 0.492·41-s + 0.800·43-s + 0.917·45-s + 1.27·47-s + 1.84·49-s + 0.477·51-s + 1.34·53-s − 0.323·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.296731915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.296731915\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.658T + 3T^{2} \) |
| 5 | \( 1 + 2.39T + 5T^{2} \) |
| 7 | \( 1 - 4.45T + 7T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 + 5.17T + 17T^{2} \) |
| 23 | \( 1 - 0.851T + 23T^{2} \) |
| 29 | \( 1 - 7.62T + 29T^{2} \) |
| 31 | \( 1 - 6.34T + 31T^{2} \) |
| 37 | \( 1 - 9.00T + 37T^{2} \) |
| 41 | \( 1 + 3.15T + 41T^{2} \) |
| 43 | \( 1 - 5.24T + 43T^{2} \) |
| 47 | \( 1 - 8.77T + 47T^{2} \) |
| 53 | \( 1 - 9.81T + 53T^{2} \) |
| 59 | \( 1 + 1.57T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 3.43T + 67T^{2} \) |
| 71 | \( 1 + 4.02T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 + 4.15T + 79T^{2} \) |
| 83 | \( 1 - 6.27T + 83T^{2} \) |
| 89 | \( 1 - 1.22T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−10.69825728765251849222743679639, −9.073080538782031746661931974751, −8.339259910740620050934515774841, −7.928507566024172970074254379900, −6.76237944348761582930208432220, −5.77843893104613163556827997534, −4.67551708475743204009967841728, −4.10473091901220930465494172079, −2.61092831636976701271781426758, −0.987068161358979526923613269031,
0.987068161358979526923613269031, 2.61092831636976701271781426758, 4.10473091901220930465494172079, 4.67551708475743204009967841728, 5.77843893104613163556827997534, 6.76237944348761582930208432220, 7.928507566024172970074254379900, 8.339259910740620050934515774841, 9.073080538782031746661931974751, 10.69825728765251849222743679639
