L(s) = 1 | + (1.5 + 0.866i)3-s + (1 − 1.73i)4-s + (−0.5 + 0.866i)5-s + (2 + 3.46i)7-s + (1.5 + 2.59i)9-s + (3 − 1.73i)12-s + (−0.5 + 0.866i)13-s + (−1.5 + 0.866i)15-s + (−1.99 − 3.46i)16-s − 3·17-s + 2·19-s + (0.999 + 1.73i)20-s + 6.92i·21-s + (−1.5 + 2.59i)23-s + (−0.499 − 0.866i)25-s + ⋯ |
L(s) = 1 | + (0.866 + 0.499i)3-s + (0.5 − 0.866i)4-s + (−0.223 + 0.387i)5-s + (0.755 + 1.30i)7-s + (0.5 + 0.866i)9-s + (0.866 − 0.499i)12-s + (−0.138 + 0.240i)13-s + (−0.387 + 0.223i)15-s + (−0.499 − 0.866i)16-s − 0.727·17-s + 0.458·19-s + (0.223 + 0.387i)20-s + 1.51i·21-s + (−0.312 + 0.541i)23-s + (−0.0999 − 0.173i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.07745 + 0.756133i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.07745 + 0.756133i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + (-6 + 10.3i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 + (-3.5 - 6.06i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.93010050190578131709217144376, −9.689129838010136175015076660757, −9.272857555949568460027828887445, −8.187721209594576722270502240898, −7.40009806758018851346513081487, −6.13053430042132171226545804344, −5.26508227904351933532283758267, −4.20687796837548792967808654514, −2.65663678564189498846180835172, −1.97082098274164565420682191991,
1.30735363288778794687760949473, 2.70857127280437685586188788549, 3.84824401142287842985131840707, 4.62121900128406685981824770763, 6.46801139844877413932153507920, 7.32942057489335448414199797032, 7.87536326563363653993875262780, 8.531114317761551315792798865620, 9.576769722926216955152214017708, 10.78109804584532360233798481237