| L(s) = 1 | + 3-s + 5-s + 9-s + 3.64·11-s + 2.64·13-s + 15-s + 3.64·17-s − 2.29·19-s − 3.64·23-s + 25-s + 27-s + 2.35·29-s + 6.29·31-s + 3.64·33-s − 4.64·37-s + 2.64·39-s − 10.9·41-s + 9.93·43-s + 45-s + 6·47-s + 3.64·51-s − 7.29·53-s + 3.64·55-s − 2.29·57-s − 4.93·59-s + 8·61-s + 2.64·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.333·9-s + 1.09·11-s + 0.733·13-s + 0.258·15-s + 0.884·17-s − 0.525·19-s − 0.760·23-s + 0.200·25-s + 0.192·27-s + 0.437·29-s + 1.12·31-s + 0.634·33-s − 0.763·37-s + 0.423·39-s − 1.70·41-s + 1.51·43-s + 0.149·45-s + 0.875·47-s + 0.510·51-s − 1.00·53-s + 0.491·55-s − 0.303·57-s − 0.642·59-s + 1.02·61-s + 0.328·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.891660135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.891660135\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 3.64T + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 - 3.64T + 17T^{2} \) |
| 19 | \( 1 + 2.29T + 19T^{2} \) |
| 23 | \( 1 + 3.64T + 23T^{2} \) |
| 29 | \( 1 - 2.35T + 29T^{2} \) |
| 31 | \( 1 - 6.29T + 31T^{2} \) |
| 37 | \( 1 + 4.64T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 9.93T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 7.29T + 53T^{2} \) |
| 59 | \( 1 + 4.93T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 4.64T + 67T^{2} \) |
| 71 | \( 1 - 8.35T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 8.29T + 79T^{2} \) |
| 83 | \( 1 - 4.93T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.730249532677130287701797653528, −8.176017037204399438855129628137, −7.25603280827729751084484723326, −6.40922487170392511380288402038, −5.89305144662455298053584140657, −4.75377636089594748705840837704, −3.90114258681527509345805358463, −3.16440206342197677931664617320, −2.02518995838370549514561001151, −1.10818831408524483404033384072,
1.10818831408524483404033384072, 2.02518995838370549514561001151, 3.16440206342197677931664617320, 3.90114258681527509345805358463, 4.75377636089594748705840837704, 5.89305144662455298053584140657, 6.40922487170392511380288402038, 7.25603280827729751084484723326, 8.176017037204399438855129628137, 8.730249532677130287701797653528
