| L(s) = 1 | + 9.63·3-s + 5.91·5-s + 65.8·9-s + 18.3·11-s − 90.9·13-s + 57.0·15-s + 91.9·17-s + 124.·19-s − 65.0·23-s − 89.9·25-s + 374.·27-s + 83.4·29-s + 165.·31-s + 177.·33-s + 310.·37-s − 876.·39-s + 181.·41-s − 82.3·43-s + 389.·45-s − 367.·47-s + 885.·51-s + 52.6·53-s + 108.·55-s + 1.19e3·57-s − 561.·59-s + 276.·61-s − 538.·65-s + ⋯ |
| L(s) = 1 | + 1.85·3-s + 0.529·5-s + 2.44·9-s + 0.503·11-s − 1.94·13-s + 0.981·15-s + 1.31·17-s + 1.50·19-s − 0.589·23-s − 0.719·25-s + 2.67·27-s + 0.534·29-s + 0.957·31-s + 0.933·33-s + 1.38·37-s − 3.60·39-s + 0.691·41-s − 0.292·43-s + 1.29·45-s − 1.14·47-s + 2.43·51-s + 0.136·53-s + 0.266·55-s + 2.78·57-s − 1.23·59-s + 0.579·61-s − 1.02·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.827059255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.827059255\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 9.63T + 27T^{2} \) |
| 5 | \( 1 - 5.91T + 125T^{2} \) |
| 11 | \( 1 - 18.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 90.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 91.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 124.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 65.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 83.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 310.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 181.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 82.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 367.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 52.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 561.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 276.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 60.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 155.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 731.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 290.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 43.5T + 5.71e5T^{2} \) |
| 89 | \( 1 + 393.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 521.T + 9.12e5T^{2} \) |
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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
−9.689417624946326628657432466885, −9.340874520523949061247282226911, −7.892467118003388220251855094313, −7.79778017600795823277505413376, −6.68638893680957192008339725506, −5.29730521991112322521795846649, −4.22708408556563881367061939228, −3.11046096880991467355778484612, −2.42717953890251099754563115722, −1.27180874924578794722909264041,
1.27180874924578794722909264041, 2.42717953890251099754563115722, 3.11046096880991467355778484612, 4.22708408556563881367061939228, 5.29730521991112322521795846649, 6.68638893680957192008339725506, 7.79778017600795823277505413376, 7.892467118003388220251855094313, 9.340874520523949061247282226911, 9.689417624946326628657432466885
