| L(s) = 1 | − 9·3-s − 70.9·5-s + 28.4·7-s + 81·9-s − 594.·11-s − 169·13-s + 638.·15-s + 971.·17-s − 1.66e3·19-s − 256.·21-s − 3.44e3·23-s + 1.91e3·25-s − 729·27-s − 964.·29-s − 7.88e3·31-s + 5.35e3·33-s − 2.02e3·35-s − 6.34e3·37-s + 1.52e3·39-s + 2.31e3·41-s − 6.40e3·43-s − 5.74e3·45-s + 1.04e4·47-s − 1.59e4·49-s − 8.74e3·51-s + 8.42e3·53-s + 4.22e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.26·5-s + 0.219·7-s + 0.333·9-s − 1.48·11-s − 0.277·13-s + 0.732·15-s + 0.815·17-s − 1.06·19-s − 0.126·21-s − 1.35·23-s + 0.611·25-s − 0.192·27-s − 0.212·29-s − 1.47·31-s + 0.855·33-s − 0.278·35-s − 0.761·37-s + 0.160·39-s + 0.215·41-s − 0.527·43-s − 0.423·45-s + 0.687·47-s − 0.951·49-s − 0.470·51-s + 0.411·53-s + 1.88·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2132840169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2132840169\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 13 | \( 1 + 169T \) |
| good | 5 | \( 1 + 70.9T + 3.12e3T^{2} \) |
| 7 | \( 1 - 28.4T + 1.68e4T^{2} \) |
| 11 | \( 1 + 594.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 971.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.66e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.44e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 964.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.88e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.34e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.31e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.40e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.04e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.42e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.10e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.61e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.54e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.19e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.85e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.31e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.29e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.48e4T + 8.58e9T^{2} \) |
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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
−10.15447239096696604575527800624, −8.813085063696632828508755520452, −7.71899694170262651233710121011, −7.58255626770784725965165423506, −6.15391024494025854576386526788, −5.20353171256444335792694479849, −4.31485966377277382886313562041, −3.31489796375953869853950228936, −1.91113225485209793703329716781, −0.21830398254140576410467392768,
0.21830398254140576410467392768, 1.91113225485209793703329716781, 3.31489796375953869853950228936, 4.31485966377277382886313562041, 5.20353171256444335792694479849, 6.15391024494025854576386526788, 7.58255626770784725965165423506, 7.71899694170262651233710121011, 8.813085063696632828508755520452, 10.15447239096696604575527800624
