| L(s) = 1 | − 2·5-s + 11-s + 13-s + 6·17-s + 4·23-s − 25-s − 6·29-s − 2·31-s + 2·37-s + 4·41-s − 10·43-s + 8·47-s − 7·49-s − 2·55-s + 4·59-s + 10·61-s − 2·65-s + 2·67-s − 8·71-s − 10·73-s − 2·79-s + 4·83-s − 12·85-s + 6·89-s − 10·97-s + 2·101-s + 20·107-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.301·11-s + 0.277·13-s + 1.45·17-s + 0.834·23-s − 1/5·25-s − 1.11·29-s − 0.359·31-s + 0.328·37-s + 0.624·41-s − 1.52·43-s + 1.16·47-s − 49-s − 0.269·55-s + 0.520·59-s + 1.28·61-s − 0.248·65-s + 0.244·67-s − 0.949·71-s − 1.17·73-s − 0.225·79-s + 0.439·83-s − 1.30·85-s + 0.635·89-s − 1.01·97-s + 0.199·101-s + 1.93·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.591414003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.591414003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.141277437287908245433419733972, −7.50550034329494737078831395309, −6.99028845543725039031640631935, −5.97020779702189898297163228526, −5.35924054969335723453614766673, −4.42380939813154139364848323539, −3.64882527929769846613790943372, −3.11026424115305134240410904985, −1.79665958160734626453048683021, −0.69902672648792210378171718758,
0.69902672648792210378171718758, 1.79665958160734626453048683021, 3.11026424115305134240410904985, 3.64882527929769846613790943372, 4.42380939813154139364848323539, 5.35924054969335723453614766673, 5.97020779702189898297163228526, 6.99028845543725039031640631935, 7.50550034329494737078831395309, 8.141277437287908245433419733972
