| L(s) = 1 | − 1.15·3-s − 4.09·5-s − 0.621·7-s − 1.66·9-s + 0.176·11-s − 6.62·13-s + 4.73·15-s − 4.87·17-s − 6.67·19-s + 0.718·21-s + 1.95·23-s + 11.7·25-s + 5.39·27-s − 4.79·29-s − 3.71·31-s − 0.204·33-s + 2.54·35-s + 6.22·37-s + 7.65·39-s + 1.84·41-s − 6.89·43-s + 6.81·45-s − 0.545·47-s − 6.61·49-s + 5.63·51-s − 4.30·53-s − 0.723·55-s + ⋯ |
| L(s) = 1 | − 0.667·3-s − 1.83·5-s − 0.234·7-s − 0.554·9-s + 0.0532·11-s − 1.83·13-s + 1.22·15-s − 1.18·17-s − 1.53·19-s + 0.156·21-s + 0.406·23-s + 2.35·25-s + 1.03·27-s − 0.891·29-s − 0.666·31-s − 0.0355·33-s + 0.430·35-s + 1.02·37-s + 1.22·39-s + 0.288·41-s − 1.05·43-s + 1.01·45-s − 0.0795·47-s − 0.944·49-s + 0.789·51-s − 0.591·53-s − 0.0975·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1093735802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1093735802\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 1.15T + 3T^{2} \) |
| 5 | \( 1 + 4.09T + 5T^{2} \) |
| 7 | \( 1 + 0.621T + 7T^{2} \) |
| 11 | \( 1 - 0.176T + 11T^{2} \) |
| 13 | \( 1 + 6.62T + 13T^{2} \) |
| 17 | \( 1 + 4.87T + 17T^{2} \) |
| 19 | \( 1 + 6.67T + 19T^{2} \) |
| 23 | \( 1 - 1.95T + 23T^{2} \) |
| 29 | \( 1 + 4.79T + 29T^{2} \) |
| 31 | \( 1 + 3.71T + 31T^{2} \) |
| 37 | \( 1 - 6.22T + 37T^{2} \) |
| 41 | \( 1 - 1.84T + 41T^{2} \) |
| 43 | \( 1 + 6.89T + 43T^{2} \) |
| 47 | \( 1 + 0.545T + 47T^{2} \) |
| 53 | \( 1 + 4.30T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 3.06T + 61T^{2} \) |
| 67 | \( 1 + 9.29T + 67T^{2} \) |
| 71 | \( 1 + 6.86T + 71T^{2} \) |
| 73 | \( 1 - 5.09T + 73T^{2} \) |
| 79 | \( 1 + 7.28T + 79T^{2} \) |
| 83 | \( 1 + 18.1T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 8.30T + 97T^{2} \) |
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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
−8.950425955387767669559512882637, −8.378088306430258283126196300801, −7.46104236751796552298562215354, −6.95590106901259628903054010763, −6.06414185838434235636572019618, −4.80904844242356679898289286043, −4.48417042190867064088588800570, −3.38216127980275286911506228262, −2.35639967432408656275921486253, −0.20993773650123274223477480162,
0.20993773650123274223477480162, 2.35639967432408656275921486253, 3.38216127980275286911506228262, 4.48417042190867064088588800570, 4.80904844242356679898289286043, 6.06414185838434235636572019618, 6.95590106901259628903054010763, 7.46104236751796552298562215354, 8.378088306430258283126196300801, 8.950425955387767669559512882637
