| L(s) = 1 | + 2.04·3-s + 7.01·5-s + 7·7-s − 22.8·9-s + 59.9·11-s + 66.4·13-s + 14.3·15-s + 26.8·17-s + 115.·19-s + 14.3·21-s − 27.5·23-s − 75.7·25-s − 101.·27-s + 134.·29-s − 263.·31-s + 122.·33-s + 49.1·35-s − 99.6·37-s + 136.·39-s − 41·41-s − 103.·43-s − 160.·45-s + 140.·47-s + 49·49-s + 54.9·51-s + 379.·53-s + 420.·55-s + ⋯ |
| L(s) = 1 | + 0.393·3-s + 0.627·5-s + 0.377·7-s − 0.844·9-s + 1.64·11-s + 1.41·13-s + 0.247·15-s + 0.383·17-s + 1.39·19-s + 0.148·21-s − 0.249·23-s − 0.606·25-s − 0.726·27-s + 0.858·29-s − 1.52·31-s + 0.646·33-s + 0.237·35-s − 0.442·37-s + 0.558·39-s − 0.156·41-s − 0.366·43-s − 0.530·45-s + 0.435·47-s + 0.142·49-s + 0.150·51-s + 0.982·53-s + 1.03·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.456296627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.456296627\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 41 | \( 1 + 41T \) |
| good | 3 | \( 1 - 2.04T + 27T^{2} \) |
| 5 | \( 1 - 7.01T + 125T^{2} \) |
| 11 | \( 1 - 59.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 26.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 115.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 27.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 134.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 263.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 99.6T + 5.06e4T^{2} \) |
| 43 | \( 1 + 103.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 140.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 379.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 281.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 201.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 333.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 292.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 638.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 275.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 669.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 884.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.07e3T + 9.12e5T^{2} \) |
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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
−9.242985685703849680709855399045, −8.772209041582122841963252761991, −7.933269770170585394900384115057, −6.86115172524380779363842325274, −5.97392665469752054484946667625, −5.38048911160253163515832349578, −3.92921630954994348224872661868, −3.29925168341589575469914816830, −1.90699476852567147780451459443, −1.02971724641968116101659444039,
1.02971724641968116101659444039, 1.90699476852567147780451459443, 3.29925168341589575469914816830, 3.92921630954994348224872661868, 5.38048911160253163515832349578, 5.97392665469752054484946667625, 6.86115172524380779363842325274, 7.933269770170585394900384115057, 8.772209041582122841963252761991, 9.242985685703849680709855399045
