| L(s) = 1 | − 13.9·3-s + 25·5-s − 49·7-s − 48.2·9-s + 473.·11-s + 22.8·13-s − 348.·15-s − 1.78e3·17-s + 2.95e3·19-s + 683.·21-s − 3.33e3·23-s + 625·25-s + 4.06e3·27-s − 4.71e3·29-s + 1.59e3·31-s − 6.60e3·33-s − 1.22e3·35-s + 7.78e3·37-s − 319.·39-s − 5.78e3·41-s + 4.47e3·43-s − 1.20e3·45-s − 2.95e4·47-s + 2.40e3·49-s + 2.48e4·51-s − 2.43e4·53-s + 1.18e4·55-s + ⋯ |
| L(s) = 1 | − 0.895·3-s + 0.447·5-s − 0.377·7-s − 0.198·9-s + 1.17·11-s + 0.0375·13-s − 0.400·15-s − 1.49·17-s + 1.87·19-s + 0.338·21-s − 1.31·23-s + 0.200·25-s + 1.07·27-s − 1.04·29-s + 0.297·31-s − 1.05·33-s − 0.169·35-s + 0.935·37-s − 0.0336·39-s − 0.537·41-s + 0.369·43-s − 0.0887·45-s − 1.95·47-s + 0.142·49-s + 1.33·51-s − 1.18·53-s + 0.527·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.307140315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.307140315\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 13.9T + 243T^{2} \) |
| 11 | \( 1 - 473.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 22.8T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.78e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.95e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.33e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.78e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.78e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.47e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.95e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.58e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.03e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.50e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.66e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.52e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.69e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.57e4T + 8.58e9T^{2} \) |
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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
−9.838403647224752258792845801438, −9.364413189812350444241898496264, −8.267331005106417114691119282323, −6.96453507312492179000808193534, −6.27906763877028776629715474808, −5.54311933706446133915203558656, −4.46048202318699176311695645973, −3.26737492078558036779971433646, −1.84697395585055937210570550532, −0.58813053924273579626119295163,
0.58813053924273579626119295163, 1.84697395585055937210570550532, 3.26737492078558036779971433646, 4.46048202318699176311695645973, 5.54311933706446133915203558656, 6.27906763877028776629715474808, 6.96453507312492179000808193534, 8.267331005106417114691119282323, 9.364413189812350444241898496264, 9.838403647224752258792845801438
