| L(s) = 1 | + 0.261·3-s + 19.5·5-s − 26.9·9-s + 56.2·11-s − 66.0·13-s + 5.10·15-s − 18.8·17-s + 80.8·19-s + 89.3·23-s + 255.·25-s − 14.1·27-s + 104.·29-s + 148.·31-s + 14.7·33-s + 13.8·37-s − 17.2·39-s + 174.·41-s − 205.·43-s − 525.·45-s + 116.·47-s − 4.92·51-s − 316.·53-s + 1.09e3·55-s + 21.1·57-s + 539.·59-s + 145.·61-s − 1.28e3·65-s + ⋯ |
| L(s) = 1 | + 0.0503·3-s + 1.74·5-s − 0.997·9-s + 1.54·11-s − 1.40·13-s + 0.0877·15-s − 0.268·17-s + 0.975·19-s + 0.810·23-s + 2.04·25-s − 0.100·27-s + 0.666·29-s + 0.860·31-s + 0.0776·33-s + 0.0613·37-s − 0.0709·39-s + 0.664·41-s − 0.728·43-s − 1.73·45-s + 0.361·47-s − 0.0135·51-s − 0.819·53-s + 2.69·55-s + 0.0491·57-s + 1.19·59-s + 0.305·61-s − 2.45·65-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
| \(L(2)\) |
\(\approx\) |
\(2.987915084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.987915084\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 0.261T + 27T^{2} \) |
| 5 | \( 1 - 19.5T + 125T^{2} \) |
| 11 | \( 1 - 56.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 66.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 80.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 89.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 104.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 148.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 13.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 174.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 205.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 116.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 316.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 539.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 145.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 476.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 131.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 349.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 806.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 233.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 535.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 80.7T + 9.12e5T^{2} \) |
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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
−9.558119634309433209158688917523, −9.406026634629488325437614520535, −8.425923223015454020672843616901, −7.05125234328107782356143058685, −6.36437418520079425534683407721, −5.51814401869321549884248137248, −4.70805751365545198997272390060, −3.09116555443064078350948116877, −2.22294881471737249996642953451, −1.01850049181261378113776757309,
1.01850049181261378113776757309, 2.22294881471737249996642953451, 3.09116555443064078350948116877, 4.70805751365545198997272390060, 5.51814401869321549884248137248, 6.36437418520079425534683407721, 7.05125234328107782356143058685, 8.425923223015454020672843616901, 9.406026634629488325437614520535, 9.558119634309433209158688917523
