| L(s) = 1 | − 7.94·3-s + 0.0633·5-s − 19.8·7-s + 36.1·9-s + 64.7·11-s − 32.5·13-s − 0.503·15-s − 95.7·17-s + 19·19-s + 157.·21-s + 160.·23-s − 124.·25-s − 73.1·27-s + 34.1·29-s − 212.·31-s − 514.·33-s − 1.25·35-s − 330.·37-s + 258.·39-s − 238.·41-s + 469.·43-s + 2.29·45-s − 354.·47-s + 50.1·49-s + 761.·51-s − 489.·53-s + 4.10·55-s + ⋯ |
| L(s) = 1 | − 1.52·3-s + 0.00566·5-s − 1.07·7-s + 1.34·9-s + 1.77·11-s − 0.693·13-s − 0.00866·15-s − 1.36·17-s + 0.229·19-s + 1.63·21-s + 1.45·23-s − 0.999·25-s − 0.521·27-s + 0.218·29-s − 1.23·31-s − 2.71·33-s − 0.00606·35-s − 1.46·37-s + 1.06·39-s − 0.910·41-s + 1.66·43-s + 0.00759·45-s − 1.10·47-s + 0.146·49-s + 2.08·51-s − 1.26·53-s + 0.0100·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6051305904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6051305904\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 + 7.94T + 27T^{2} \) |
| 5 | \( 1 - 0.0633T + 125T^{2} \) |
| 7 | \( 1 + 19.8T + 343T^{2} \) |
| 11 | \( 1 - 64.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 32.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 95.7T + 4.91e3T^{2} \) |
| 23 | \( 1 - 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 34.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 330.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 238.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 469.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 354.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 489.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 385.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 859.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 454.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 445.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 598.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 107.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 692.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 143.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.34e3T + 9.12e5T^{2} \) |
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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
−9.406948492522994410206881658529, −8.879119936314448481279468525356, −7.21390865123879452129114870555, −6.70996529765837603235585288987, −6.17034502504427172163000605681, −5.19718461968127760415139203209, −4.32391427542435304640792236458, −3.32853377814071528929816701540, −1.70798381361036475998275562206, −0.43115429723847000823710233320,
0.43115429723847000823710233320, 1.70798381361036475998275562206, 3.32853377814071528929816701540, 4.32391427542435304640792236458, 5.19718461968127760415139203209, 6.17034502504427172163000605681, 6.70996529765837603235585288987, 7.21390865123879452129114870555, 8.879119936314448481279468525356, 9.406948492522994410206881658529
