| L(s) = 1 | − 2.82·5-s + 26·11-s + 33.9·13-s + 103.·17-s + 94.7·19-s + 148·23-s − 117·25-s + 118·29-s − 296.·31-s − 254·37-s − 91.9·41-s + 122·43-s − 308.·47-s + 170·53-s − 73.5·55-s − 304.·59-s − 608.·61-s − 96·65-s + 420·67-s − 420·71-s + 813.·73-s + 1.05e3·79-s + 1.44e3·83-s − 292·85-s + 1.02e3·89-s − 268·95-s + 315.·97-s + ⋯ |
| L(s) = 1 | − 0.252·5-s + 0.712·11-s + 0.724·13-s + 1.47·17-s + 1.14·19-s + 1.34·23-s − 0.936·25-s + 0.755·29-s − 1.72·31-s − 1.12·37-s − 0.350·41-s + 0.432·43-s − 0.956·47-s + 0.440·53-s − 0.180·55-s − 0.670·59-s − 1.27·61-s − 0.183·65-s + 0.765·67-s − 0.702·71-s + 1.30·73-s + 1.49·79-s + 1.91·83-s − 0.372·85-s + 1.22·89-s − 0.289·95-s + 0.330·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.508990895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.508990895\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.82T + 125T^{2} \) |
| 11 | \( 1 - 26T + 1.33e3T^{2} \) |
| 13 | \( 1 - 33.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 94.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 148T + 1.21e4T^{2} \) |
| 29 | \( 1 - 118T + 2.43e4T^{2} \) |
| 31 | \( 1 + 296.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 254T + 5.06e4T^{2} \) |
| 41 | \( 1 + 91.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 122T + 7.95e4T^{2} \) |
| 47 | \( 1 + 308.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 170T + 1.48e5T^{2} \) |
| 59 | \( 1 + 304.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 608.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 420T + 3.00e5T^{2} \) |
| 71 | \( 1 + 420T + 3.57e5T^{2} \) |
| 73 | \( 1 - 813.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 315.T + 9.12e5T^{2} \) |
| 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
−9.065985864566275333021450701743, −8.065547437942526271104764914135, −7.42026536791943421014216345588, −6.59011688994847988801252218784, −5.64395622996043579273944067524, −4.95196028376628184768375581884, −3.64811408218311711266126416946, −3.27349222165842793856606665438, −1.68821923170462686005375469496, −0.799405893773458280955515253118,
0.799405893773458280955515253118, 1.68821923170462686005375469496, 3.27349222165842793856606665438, 3.64811408218311711266126416946, 4.95196028376628184768375581884, 5.64395622996043579273944067524, 6.59011688994847988801252218784, 7.42026536791943421014216345588, 8.065547437942526271104764914135, 9.065985864566275333021450701743
