| L(s) = 1 | − 23.5·3-s − 74.2·5-s + 310.·9-s + 424.·11-s − 252.·13-s + 1.74e3·15-s − 1.10e3·17-s + 6.47·19-s + 3.61e3·23-s + 2.39e3·25-s − 1.57e3·27-s − 5.00e3·29-s − 2.82e3·31-s − 9.97e3·33-s − 2.04e3·37-s + 5.93e3·39-s + 9.39e3·41-s − 1.03e4·43-s − 2.30e4·45-s − 1.70e4·47-s + 2.59e4·51-s − 3.95e4·53-s − 3.15e4·55-s − 152.·57-s − 3.39e4·59-s + 2.82e4·61-s + 1.87e4·65-s + ⋯ |
| L(s) = 1 | − 1.50·3-s − 1.32·5-s + 1.27·9-s + 1.05·11-s − 0.413·13-s + 2.00·15-s − 0.926·17-s + 0.00411·19-s + 1.42·23-s + 0.765·25-s − 0.416·27-s − 1.10·29-s − 0.527·31-s − 1.59·33-s − 0.245·37-s + 0.624·39-s + 0.872·41-s − 0.851·43-s − 1.69·45-s − 1.12·47-s + 1.39·51-s − 1.93·53-s − 1.40·55-s − 0.00620·57-s − 1.26·59-s + 0.973·61-s + 0.550·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3500205395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3500205395\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 23.5T + 243T^{2} \) |
| 5 | \( 1 + 74.2T + 3.12e3T^{2} \) |
| 11 | \( 1 - 424.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 252.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.10e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 6.47T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.61e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.04e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.70e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.95e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.39e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.55e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.82e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.38e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.88e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.08e5T + 8.58e9T^{2} \) |
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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
−9.574895245928664270000755873698, −8.729105255502579034466226908487, −7.53506403805453688359242026333, −6.89428524545421763848815626830, −6.10999532522848571207363246187, −4.95530097879747182709761881640, −4.34782564733914774789390977894, −3.30770949362189542206696523423, −1.48371166706497538348399988319, −0.30864521820180407176553164273,
0.30864521820180407176553164273, 1.48371166706497538348399988319, 3.30770949362189542206696523423, 4.34782564733914774789390977894, 4.95530097879747182709761881640, 6.10999532522848571207363246187, 6.89428524545421763848815626830, 7.53506403805453688359242026333, 8.729105255502579034466226908487, 9.574895245928664270000755873698
