| L(s) = 1 | − 4·2-s − 22.6·3-s + 16·4-s + 79.6·5-s + 90.5·6-s + 191.·7-s − 64·8-s + 269.·9-s − 318.·10-s − 358.·11-s − 362.·12-s + 443.·13-s − 767.·14-s − 1.80e3·15-s + 256·16-s + 991.·17-s − 1.07e3·18-s + 1.27e3·20-s − 4.34e3·21-s + 1.43e3·22-s + 1.33e3·23-s + 1.44e3·24-s + 3.22e3·25-s − 1.77e3·26-s − 608.·27-s + 3.06e3·28-s + 5.07e3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.45·3-s + 0.5·4-s + 1.42·5-s + 1.02·6-s + 1.47·7-s − 0.353·8-s + 1.11·9-s − 1.00·10-s − 0.892·11-s − 0.726·12-s + 0.728·13-s − 1.04·14-s − 2.07·15-s + 0.250·16-s + 0.832·17-s − 0.785·18-s + 0.712·20-s − 2.14·21-s + 0.631·22-s + 0.527·23-s + 0.513·24-s + 1.03·25-s − 0.514·26-s − 0.160·27-s + 0.739·28-s + 1.12·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.745131284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.745131284\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 22.6T + 243T^{2} \) |
| 5 | \( 1 - 79.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 191.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 358.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 443.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 991.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.33e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.07e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.32e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.88e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.12e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.81e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.06e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.60e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.21e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.65e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.08e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.05e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.00e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.09e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.17e4T + 8.58e9T^{2} \) |
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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
−9.944182140786725205947519898329, −8.804002152949952277918090467021, −7.971311903087846288076703718249, −6.90970151583808664449753279881, −5.90523071267449487205152939397, −5.46259705046085783293017985951, −4.64300089501306553437445283409, −2.61630040057563386502044547799, −1.46448769072775217270886780521, −0.833757968037833567425451947549,
0.833757968037833567425451947549, 1.46448769072775217270886780521, 2.61630040057563386502044547799, 4.64300089501306553437445283409, 5.46259705046085783293017985951, 5.90523071267449487205152939397, 6.90970151583808664449753279881, 7.971311903087846288076703718249, 8.804002152949952277918090467021, 9.944182140786725205947519898329
