| L(s) = 1 | + 25·5-s + 80.4·7-s − 520.·11-s + 421.·13-s + 1.16e3·17-s − 1.17e3·19-s − 2.37e3·23-s + 625·25-s − 7.05e3·29-s + 5.56e3·31-s + 2.01e3·35-s + 2.38e3·37-s + 8.38e3·41-s − 2.05e4·43-s − 8.35e3·47-s − 1.03e4·49-s + 2.83e4·53-s − 1.30e4·55-s − 3.75e4·59-s − 5.73e4·61-s + 1.05e4·65-s + 4.58e4·67-s − 4.15e4·71-s + 7.68e4·73-s − 4.18e4·77-s + 7.00e3·79-s + 9.32e4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.620·7-s − 1.29·11-s + 0.692·13-s + 0.975·17-s − 0.743·19-s − 0.935·23-s + 0.200·25-s − 1.55·29-s + 1.03·31-s + 0.277·35-s + 0.287·37-s + 0.779·41-s − 1.69·43-s − 0.551·47-s − 0.615·49-s + 1.38·53-s − 0.579·55-s − 1.40·59-s − 1.97·61-s + 0.309·65-s + 1.24·67-s − 0.977·71-s + 1.68·73-s − 0.804·77-s + 0.126·79-s + 1.48·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 7 | \( 1 - 80.4T + 1.68e4T^{2} \) |
| 11 | \( 1 + 520.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 421.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.16e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.37e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.05e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.38e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.35e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.83e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.73e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.58e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.68e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.00e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.32e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.07e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.24e4T + 8.58e9T^{2} \) |
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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
−9.264022381566053181180844550577, −8.119535516240409285912243990331, −7.79693037668850581504546382338, −6.42602856011680834472000288017, −5.61154397095213925894312707784, −4.79548307726262711202326408146, −3.59014143792915544706668369893, −2.39796887168296471111183148469, −1.41649065660301200567876111711, 0,
1.41649065660301200567876111711, 2.39796887168296471111183148469, 3.59014143792915544706668369893, 4.79548307726262711202326408146, 5.61154397095213925894312707784, 6.42602856011680834472000288017, 7.79693037668850581504546382338, 8.119535516240409285912243990331, 9.264022381566053181180844550577
