| L(s) = 1 | + 4.31·2-s + 5·3-s + 10.6·4-s + 5·5-s + 21.5·6-s + 11.3·8-s − 2·9-s + 21.5·10-s + 19.7·11-s + 53.1·12-s + 71.3·13-s + 25·15-s − 36.0·16-s − 31.3·17-s − 8.63·18-s + 136.·19-s + 53.1·20-s + 85.1·22-s − 100.·23-s + 56.8·24-s + 25·25-s + 307.·26-s − 145·27-s − 288.·29-s + 107.·30-s − 208.·31-s − 246.·32-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 0.962·3-s + 1.32·4-s + 0.447·5-s + 1.46·6-s + 0.502·8-s − 0.0740·9-s + 0.682·10-s + 0.540·11-s + 1.27·12-s + 1.52·13-s + 0.430·15-s − 0.562·16-s − 0.447·17-s − 0.113·18-s + 1.64·19-s + 0.594·20-s + 0.825·22-s − 0.914·23-s + 0.483·24-s + 0.200·25-s + 2.32·26-s − 1.03·27-s − 1.84·29-s + 0.656·30-s − 1.21·31-s − 1.36·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.759021610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.759021610\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 4.31T + 8T^{2} \) |
| 3 | \( 1 - 5T + 27T^{2} \) |
| 11 | \( 1 - 19.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 71.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 31.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 136.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 100.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 288.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 208.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 309.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 181.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 18.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 127.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 322.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 341.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 84.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 315.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.23e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 643.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.41e3T + 9.12e5T^{2} \) |
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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
−11.74587977499222614555230142234, −11.15330591836305629358613737650, −9.540686116183285924535022904666, −8.804817775980628669730559274887, −7.51979363449750788782525662000, −6.19995531418782793662909315371, −5.46802156274272187664260207126, −3.92591194441171380100827390481, −3.27028563145929960322759286153, −1.88561817777381399392277858905,
1.88561817777381399392277858905, 3.27028563145929960322759286153, 3.92591194441171380100827390481, 5.46802156274272187664260207126, 6.19995531418782793662909315371, 7.51979363449750788782525662000, 8.804817775980628669730559274887, 9.540686116183285924535022904666, 11.15330591836305629358613737650, 11.74587977499222614555230142234
