| L(s) = 1 | + 19·13-s − 107·19-s − 125·25-s + 289·31-s + 323·37-s + 71·43-s − 182·61-s − 127·67-s + 271·73-s − 1.38e3·79-s + 1.33e3·97-s + 1.80e3·103-s + 2.21e3·109-s + ⋯ |
| L(s) = 1 | + 0.405·13-s − 1.29·19-s − 25-s + 1.67·31-s + 1.43·37-s + 0.251·43-s − 0.382·61-s − 0.231·67-s + 0.434·73-s − 1.97·79-s + 1.39·97-s + 1.72·103-s + 1.94·109-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
| \(L(2)\) |
\(\approx\) |
\(1.942388540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.942388540\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 19 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + 107 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 - 289 T + p^{3} T^{2} \) |
| 37 | \( 1 - 323 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 - 71 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + 182 T + p^{3} T^{2} \) |
| 67 | \( 1 + 127 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 - 271 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1387 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 - 1330 T + p^{3} T^{2} \) |
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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
−8.841288018951973253271133033998, −8.203311242358778678730928983514, −7.43181219104644994514012098307, −6.36829676669733992351621948051, −5.92664242937665458075036404213, −4.68748944170006652292933682032, −4.04089509101949364659620631196, −2.90057983917499132307344325523, −1.90939280912285409944543268345, −0.65457761439550752212149809345,
0.65457761439550752212149809345, 1.90939280912285409944543268345, 2.90057983917499132307344325523, 4.04089509101949364659620631196, 4.68748944170006652292933682032, 5.92664242937665458075036404213, 6.36829676669733992351621948051, 7.43181219104644994514012098307, 8.203311242358778678730928983514, 8.841288018951973253271133033998
