| L(s) = 1 | − 3·9-s + 6·13-s − 2·17-s + 10·29-s − 2·37-s + 10·41-s − 7·49-s + 14·53-s + 10·61-s + 6·73-s + 9·81-s + 10·89-s − 18·97-s + 2·101-s − 6·109-s + 14·113-s − 18·117-s + ⋯ |
| L(s) = 1 | − 9-s + 1.66·13-s − 0.485·17-s + 1.85·29-s − 0.328·37-s + 1.56·41-s − 49-s + 1.92·53-s + 1.28·61-s + 0.702·73-s + 81-s + 1.05·89-s − 1.82·97-s + 0.199·101-s − 0.574·109-s + 1.31·113-s − 1.66·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.658334805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.658334805\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.229738681555639603454604612144, −8.548239184977679663193402766242, −8.095900450392739090116843696578, −6.83957021602347792551015258406, −6.17090310700109824442559838984, −5.42644460572577514350636284077, −4.32056015764801105587344203662, −3.38983172672051634596756490050, −2.40245494881792658011192485726, −0.923744794836951703848041902340,
0.923744794836951703848041902340, 2.40245494881792658011192485726, 3.38983172672051634596756490050, 4.32056015764801105587344203662, 5.42644460572577514350636284077, 6.17090310700109824442559838984, 6.83957021602347792551015258406, 8.095900450392739090116843696578, 8.548239184977679663193402766242, 9.229738681555639603454604612144
