| L(s) = 1 | + 3.82·2-s + 3·3-s + 6.65·4-s + 11.4·6-s + 7·7-s − 5.14·8-s + 9·9-s + 48.5·11-s + 19.9·12-s + 43.6·13-s + 26.7·14-s − 72.9·16-s + 67.6·17-s + 34.4·18-s − 93.2·19-s + 21·21-s + 185.·22-s + 104.·23-s − 15.4·24-s + 167.·26-s + 27·27-s + 46.5·28-s − 58.7·29-s − 9.08·31-s − 238.·32-s + 145.·33-s + 259.·34-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.577·3-s + 0.832·4-s + 0.781·6-s + 0.377·7-s − 0.227·8-s + 0.333·9-s + 1.33·11-s + 0.480·12-s + 0.931·13-s + 0.511·14-s − 1.13·16-s + 0.965·17-s + 0.451·18-s − 1.12·19-s + 0.218·21-s + 1.80·22-s + 0.944·23-s − 0.131·24-s + 1.26·26-s + 0.192·27-s + 0.314·28-s − 0.376·29-s − 0.0526·31-s − 1.31·32-s + 0.768·33-s + 1.30·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.521893815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.521893815\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 3.82T + 8T^{2} \) |
| 11 | \( 1 - 48.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 43.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 93.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 58.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 9.08T + 2.97e4T^{2} \) |
| 37 | \( 1 - 252.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 276.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 92.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 582.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 623.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 524.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 352.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 736.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 492.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 872.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 529.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 385.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 463.T + 9.12e5T^{2} \) |
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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
−10.76709716702433480999084431519, −9.336058746616116421305830704442, −8.793176351752138474110594018678, −7.61419417613129886177638587302, −6.47579138051612609557328574560, −5.76664371588232783169228167220, −4.45370029574047547099551712284, −3.85831518143065839884360286752, −2.80628483670701080579322390267, −1.33890609301519855356138672872,
1.33890609301519855356138672872, 2.80628483670701080579322390267, 3.85831518143065839884360286752, 4.45370029574047547099551712284, 5.76664371588232783169228167220, 6.47579138051612609557328574560, 7.61419417613129886177638587302, 8.793176351752138474110594018678, 9.336058746616116421305830704442, 10.76709716702433480999084431519
