| L(s) = 1 | − 12·3-s + 25·5-s − 49·7-s − 99·9-s + 556·11-s − 354·13-s − 300·15-s + 770·17-s − 2.68e3·19-s + 588·21-s − 1.52e3·23-s + 625·25-s + 4.10e3·27-s − 2.41e3·29-s + 7.84e3·31-s − 6.67e3·33-s − 1.22e3·35-s − 314·37-s + 4.24e3·39-s − 1.78e4·41-s + 1.64e4·43-s − 2.47e3·45-s + 5.37e3·47-s + 2.40e3·49-s − 9.24e3·51-s + 1.65e3·53-s + 1.39e4·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 0.447·5-s − 0.377·7-s − 0.407·9-s + 1.38·11-s − 0.580·13-s − 0.344·15-s + 0.646·17-s − 1.70·19-s + 0.290·21-s − 0.602·23-s + 1/5·25-s + 1.08·27-s − 0.533·29-s + 1.46·31-s − 1.06·33-s − 0.169·35-s − 0.0377·37-s + 0.447·39-s − 1.66·41-s + 1.35·43-s − 0.182·45-s + 0.354·47-s + 1/7·49-s − 0.497·51-s + 0.0808·53-s + 0.619·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.356110606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.356110606\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 + 4 p T + p^{5} T^{2} \) |
| 11 | \( 1 - 556 T + p^{5} T^{2} \) |
| 13 | \( 1 + 354 T + p^{5} T^{2} \) |
| 17 | \( 1 - 770 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2684 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1528 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2418 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7840 T + p^{5} T^{2} \) |
| 37 | \( 1 + 314 T + p^{5} T^{2} \) |
| 41 | \( 1 + 17878 T + p^{5} T^{2} \) |
| 43 | \( 1 - 16476 T + p^{5} T^{2} \) |
| 47 | \( 1 - 5376 T + p^{5} T^{2} \) |
| 53 | \( 1 - 1654 T + p^{5} T^{2} \) |
| 59 | \( 1 + 29492 T + p^{5} T^{2} \) |
| 61 | \( 1 - 27630 T + p^{5} T^{2} \) |
| 67 | \( 1 - 57716 T + p^{5} T^{2} \) |
| 71 | \( 1 - 70648 T + p^{5} T^{2} \) |
| 73 | \( 1 - 74202 T + p^{5} T^{2} \) |
| 79 | \( 1 - 74336 T + p^{5} T^{2} \) |
| 83 | \( 1 - 44068 T + p^{5} T^{2} \) |
| 89 | \( 1 - 129306 T + p^{5} T^{2} \) |
| 97 | \( 1 + 137646 T + p^{5} T^{2} \) |
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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
−11.03993691090851066650029010086, −10.12100243545431949172113643143, −9.229925736160324249474859507324, −8.197969635177208048988552475294, −6.64509786481370561227710631321, −6.20198862057844442205319402918, −5.03046683408360222816438448333, −3.77805837456073661989769635275, −2.21961225180144609024350326995, −0.68388353437272910652694347456,
0.68388353437272910652694347456, 2.21961225180144609024350326995, 3.77805837456073661989769635275, 5.03046683408360222816438448333, 6.20198862057844442205319402918, 6.64509786481370561227710631321, 8.197969635177208048988552475294, 9.229925736160324249474859507324, 10.12100243545431949172113643143, 11.03993691090851066650029010086
