| L(s) = 1 | − 1.41·5-s + 1.41·13-s + 1.41·17-s − 4·23-s − 2.99·25-s − 6·29-s + 5.65·31-s − 4·37-s + 7.07·41-s + 4·43-s − 11.3·47-s − 4·53-s + 5.65·59-s + 4.24·61-s − 2.00·65-s + 4·67-s − 12·71-s − 7.07·73-s + 8·79-s + 5.65·83-s − 2.00·85-s − 12.7·89-s + 4.24·97-s + 9.89·101-s − 11.3·103-s − 16·107-s − 4·109-s + ⋯ |
| L(s) = 1 | − 0.632·5-s + 0.392·13-s + 0.342·17-s − 0.834·23-s − 0.599·25-s − 1.11·29-s + 1.01·31-s − 0.657·37-s + 1.10·41-s + 0.609·43-s − 1.65·47-s − 0.549·53-s + 0.736·59-s + 0.543·61-s − 0.248·65-s + 0.488·67-s − 1.42·71-s − 0.827·73-s + 0.900·79-s + 0.620·83-s − 0.216·85-s − 1.34·89-s + 0.430·97-s + 0.985·101-s − 1.11·103-s − 1.54·107-s − 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 7.07T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 5.65T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
| 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
−8.023168755752908238621040280909, −7.68456555881336493880474880889, −6.70981861091610201773103587816, −5.97536612558134817180044040348, −5.18133332639539175818146219211, −4.16496564962392109862273810427, −3.63664568405146868996206917365, −2.57069469831021119584289964788, −1.41147909066713559090054133268, 0,
1.41147909066713559090054133268, 2.57069469831021119584289964788, 3.63664568405146868996206917365, 4.16496564962392109862273810427, 5.18133332639539175818146219211, 5.97536612558134817180044040348, 6.70981861091610201773103587816, 7.68456555881336493880474880889, 8.023168755752908238621040280909
