| L(s) = 1 | − 2.82·3-s + 5-s − 7-s + 5.00·9-s − 4·11-s + 13-s − 2.82·15-s + 0.828·17-s + 6.82·19-s + 2.82·21-s + 6.82·23-s + 25-s − 5.65·27-s + 6·29-s − 1.65·31-s + 11.3·33-s − 35-s − 3.65·37-s − 2.82·39-s − 10.4·41-s − 11.3·43-s + 5.00·45-s + 11.3·47-s + 49-s − 2.34·51-s + 8.82·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 1.63·3-s + 0.447·5-s − 0.377·7-s + 1.66·9-s − 1.20·11-s + 0.277·13-s − 0.730·15-s + 0.200·17-s + 1.56·19-s + 0.617·21-s + 1.42·23-s + 0.200·25-s − 1.08·27-s + 1.11·29-s − 0.297·31-s + 1.96·33-s − 0.169·35-s − 0.601·37-s − 0.452·39-s − 1.63·41-s − 1.72·43-s + 0.745·45-s + 1.65·47-s + 0.142·49-s − 0.328·51-s + 1.21·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9940917905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9940917905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 1.65T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 + 6.82T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 0.343T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 2.34T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 5.31T + 97T^{2} \) |
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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
−7.65528396808608804236421481086, −7.02673862678773194942875220204, −6.46248139657619844299389902742, −5.69829169567900237107803570424, −5.10689268017440755531100838397, −4.88480190790594557117897763258, −3.51662683075383156814036146202, −2.77644895753081758617277618141, −1.45905222776956526013392719557, −0.58510382046467137399007286378,
0.58510382046467137399007286378, 1.45905222776956526013392719557, 2.77644895753081758617277618141, 3.51662683075383156814036146202, 4.88480190790594557117897763258, 5.10689268017440755531100838397, 5.69829169567900237107803570424, 6.46248139657619844299389902742, 7.02673862678773194942875220204, 7.65528396808608804236421481086
