| L(s) = 1 | − 5-s − 5·7-s + 11-s + 5·13-s + 2·17-s + 2·19-s − 4·23-s − 4·25-s − 7·29-s − 7·31-s + 5·35-s + 6·37-s + 6·41-s + 6·43-s + 8·47-s + 18·49-s + 4·53-s − 55-s − 6·59-s − 6·61-s − 5·65-s + 5·67-s + 71-s − 4·73-s − 5·77-s − 17·79-s + 15·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.88·7-s + 0.301·11-s + 1.38·13-s + 0.485·17-s + 0.458·19-s − 0.834·23-s − 4/5·25-s − 1.29·29-s − 1.25·31-s + 0.845·35-s + 0.986·37-s + 0.937·41-s + 0.914·43-s + 1.16·47-s + 18/7·49-s + 0.549·53-s − 0.134·55-s − 0.781·59-s − 0.768·61-s − 0.620·65-s + 0.610·67-s + 0.118·71-s − 0.468·73-s − 0.569·77-s − 1.91·79-s + 1.64·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 139 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−16.71453592170783, −16.21704940202432, −15.90712212697922, −15.44985344872844, −14.63580124462090, −13.99136884442272, −13.27776894766378, −13.04387741621824, −12.28160253429498, −11.83007220945905, −11.01412421230322, −10.54986510984568, −9.697581324764620, −9.298970184380495, −8.826554083229556, −7.734452254854643, −7.444346918131601, −6.532198583572608, −5.862737660776025, −5.704707607191429, −4.149322723034036, −3.774823334415856, −3.250341354992418, −2.263991171816911, −1.036630694613953, 0,
1.036630694613953, 2.263991171816911, 3.250341354992418, 3.774823334415856, 4.149322723034036, 5.704707607191429, 5.862737660776025, 6.532198583572608, 7.444346918131601, 7.734452254854643, 8.826554083229556, 9.298970184380495, 9.697581324764620, 10.54986510984568, 11.01412421230322, 11.83007220945905, 12.28160253429498, 13.04387741621824, 13.27776894766378, 13.99136884442272, 14.63580124462090, 15.44985344872844, 15.90712212697922, 16.21704940202432, 16.71453592170783
