| L(s) = 1 | + 16.1·2-s + 71.3·3-s + 134.·4-s − 532.·5-s + 1.15e3·6-s − 6.33·7-s + 104.·8-s + 2.90e3·9-s − 8.62e3·10-s + 4.95e3·11-s + 9.59e3·12-s − 2.19e3·13-s − 102.·14-s − 3.80e4·15-s − 1.55e4·16-s + 1.51e4·17-s + 4.71e4·18-s − 8.04e3·19-s − 7.15e4·20-s − 452.·21-s + 8.02e4·22-s + 1.84e4·23-s + 7.42e3·24-s + 2.05e5·25-s − 3.55e4·26-s + 5.14e4·27-s − 851.·28-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.52·3-s + 1.05·4-s − 1.90·5-s + 2.18·6-s − 0.00698·7-s + 0.0718·8-s + 1.32·9-s − 2.72·10-s + 1.12·11-s + 1.60·12-s − 0.277·13-s − 0.00999·14-s − 2.90·15-s − 0.947·16-s + 0.748·17-s + 1.90·18-s − 0.269·19-s − 2.00·20-s − 0.0106·21-s + 1.60·22-s + 0.315·23-s + 0.109·24-s + 2.62·25-s − 0.397·26-s + 0.503·27-s − 0.00733·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.363495875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.363495875\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + 2.19e3T \) |
| good | 2 | \( 1 - 16.1T + 128T^{2} \) |
| 3 | \( 1 - 71.3T + 2.18e3T^{2} \) |
| 5 | \( 1 + 532.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 6.33T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.95e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 1.51e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 8.04e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.84e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.94e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.36e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.54e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.25e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.58e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.79e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.41e4T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.65e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.68e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.29e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.65e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.62e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.77e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.80e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.14e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.51e7T + 8.07e13T^{2} \) |
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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
−19.05377164187003266750487378504, −16.07633773662229457815017774123, −14.86762896421737596649406929776, −14.43909037725399844722488259581, −12.76576725178943671100930671529, −11.63798579468764696565977283858, −8.798523190664073435034565637355, −7.31509590067880111567405030063, −4.19901713269000046968421918571, −3.26178276460614427118453644803,
3.26178276460614427118453644803, 4.19901713269000046968421918571, 7.31509590067880111567405030063, 8.798523190664073435034565637355, 11.63798579468764696565977283858, 12.76576725178943671100930671529, 14.43909037725399844722488259581, 14.86762896421737596649406929776, 16.07633773662229457815017774123, 19.05377164187003266750487378504
