| L(s) = 1 | − 2.23·2-s + 2.23·3-s + 3.00·4-s + 3·5-s − 5.00·6-s + 2·7-s − 2.23·8-s + 2.00·9-s − 6.70·10-s − 2.23·11-s + 6.70·12-s + 13-s − 4.47·14-s + 6.70·15-s − 0.999·16-s + 4.47·17-s − 4.47·18-s + 9.00·20-s + 4.47·21-s + 5.00·22-s + 6·23-s − 5.00·24-s + 4·25-s − 2.23·26-s − 2.23·27-s + 6.00·28-s − 15.0·30-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.29·3-s + 1.50·4-s + 1.34·5-s − 2.04·6-s + 0.755·7-s − 0.790·8-s + 0.666·9-s − 2.12·10-s − 0.674·11-s + 1.93·12-s + 0.277·13-s − 1.19·14-s + 1.73·15-s − 0.249·16-s + 1.08·17-s − 1.05·18-s + 2.01·20-s + 0.975·21-s + 1.06·22-s + 1.25·23-s − 1.02·24-s + 0.800·25-s − 0.438·26-s − 0.430·27-s + 1.13·28-s − 2.73·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.536167094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.536167094\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 - 3T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 6.70T + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−9.926920400182306589509362368740, −9.166865567924464617775079142227, −8.773784570859527236552032454617, −7.83366100252561220695732628930, −7.38079612617736555181599715454, −6.05019020437305618769285639668, −4.98650870253610354787628103592, −3.18750172674517929202211975072, −2.18597365151004071826453503803, −1.39949883028894599083847629353,
1.39949883028894599083847629353, 2.18597365151004071826453503803, 3.18750172674517929202211975072, 4.98650870253610354787628103592, 6.05019020437305618769285639668, 7.38079612617736555181599715454, 7.83366100252561220695732628930, 8.773784570859527236552032454617, 9.166865567924464617775079142227, 9.926920400182306589509362368740
