| L(s) = 1 | + 64·4-s + 111·5-s − 565·7-s + 729·9-s − 813·11-s + 4.25e3·13-s + 4.09e3·16-s + 7.10e3·20-s − 3.30e3·25-s − 3.61e4·28-s − 2.47e4·29-s + 5.94e4·31-s − 6.27e4·35-s + 4.66e4·36-s − 7.67e3·37-s + 1.02e5·41-s − 5.20e4·44-s + 8.09e4·45-s + 2.07e5·47-s + 2.01e5·49-s + 2.72e5·52-s − 9.02e4·55-s − 4.11e5·63-s + 2.62e5·64-s + 4.72e5·65-s − 6.00e5·67-s − 3.85e5·71-s + ⋯ |
| L(s) = 1 | + 4-s + 0.887·5-s − 1.64·7-s + 9-s − 0.610·11-s + 1.93·13-s + 16-s + 0.887·20-s − 0.211·25-s − 1.64·28-s − 1.01·29-s + 1.99·31-s − 1.46·35-s + 36-s − 0.151·37-s + 1.48·41-s − 0.610·44-s + 0.887·45-s + 1.99·47-s + 1.71·49-s + 1.93·52-s − 0.542·55-s − 1.64·63-s + 64-s + 1.71·65-s − 1.99·67-s − 1.07·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.921319943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.921319943\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 139 | \( 1 + p^{3} T \) |
| good | 2 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 3 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 5 | \( 1 - 111 T + p^{6} T^{2} \) |
| 7 | \( 1 + 565 T + p^{6} T^{2} \) |
| 11 | \( 1 + 813 T + p^{6} T^{2} \) |
| 13 | \( 1 - 4255 T + p^{6} T^{2} \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( 1 + 24753 T + p^{6} T^{2} \) |
| 31 | \( 1 - 59443 T + p^{6} T^{2} \) |
| 37 | \( 1 + 7670 T + p^{6} T^{2} \) |
| 41 | \( 1 - 102258 T + p^{6} T^{2} \) |
| 43 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 47 | \( 1 - 207090 T + p^{6} T^{2} \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 67 | \( 1 + 600685 T + p^{6} T^{2} \) |
| 71 | \( 1 + 385197 T + p^{6} T^{2} \) |
| 73 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 79 | \( 1 - 204203 T + p^{6} T^{2} \) |
| 83 | \( 1 - 909915 T + p^{6} T^{2} \) |
| 89 | \( 1 - 989463 T + p^{6} T^{2} \) |
| 97 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
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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
−12.17612036132296889296312605526, −10.73558223353589494074477713829, −10.14218421483070218386802395078, −9.135733429087105014334496725869, −7.54397600910675935523720462055, −6.34743971830228698675893554089, −5.94770017895931667318726467706, −3.76201905211154576937062266149, −2.54373682196968993421089183978, −1.13134823212253574990220570925,
1.13134823212253574990220570925, 2.54373682196968993421089183978, 3.76201905211154576937062266149, 5.94770017895931667318726467706, 6.34743971830228698675893554089, 7.54397600910675935523720462055, 9.135733429087105014334496725869, 10.14218421483070218386802395078, 10.73558223353589494074477713829, 12.17612036132296889296312605526
