L(s) = 1 | + (1.36 − 0.382i)2-s − i·3-s + (1.70 − 1.04i)4-s − i·5-s + (−0.382 − 1.36i)6-s + 3.81·7-s + (1.92 − 2.07i)8-s − 9-s + (−0.382 − 1.36i)10-s + 3.29·11-s + (−1.04 − 1.70i)12-s − 6.74·13-s + (5.18 − 1.45i)14-s − 15-s + (1.83 − 3.55i)16-s − 2.05i·17-s + ⋯ |
L(s) = 1 | + (0.962 − 0.270i)2-s − 0.577i·3-s + (0.853 − 0.520i)4-s − 0.447i·5-s + (−0.156 − 0.555i)6-s + 1.44·7-s + (0.681 − 0.732i)8-s − 0.333·9-s + (−0.120 − 0.430i)10-s + 0.992·11-s + (−0.300 − 0.492i)12-s − 1.87·13-s + (1.38 − 0.389i)14-s − 0.258·15-s + (0.457 − 0.889i)16-s − 0.498i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0397 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0397 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.599269874\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.599269874\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 + 0.382i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (2.33 - 4.19i)T \) |
good | 7 | \( 1 - 3.81T + 7T^{2} \) |
| 11 | \( 1 - 3.29T + 11T^{2} \) |
| 13 | \( 1 + 6.74T + 13T^{2} \) |
| 17 | \( 1 + 2.05iT - 17T^{2} \) |
| 19 | \( 1 - 4.83T + 19T^{2} \) |
| 29 | \( 1 - 4.01T + 29T^{2} \) |
| 31 | \( 1 - 0.0146iT - 31T^{2} \) |
| 37 | \( 1 - 6.85iT - 37T^{2} \) |
| 41 | \( 1 + 4.62T + 41T^{2} \) |
| 43 | \( 1 + 4.10T + 43T^{2} \) |
| 47 | \( 1 + 4.94iT - 47T^{2} \) |
| 53 | \( 1 - 8.07iT - 53T^{2} \) |
| 59 | \( 1 + 7.38iT - 59T^{2} \) |
| 61 | \( 1 - 2.85iT - 61T^{2} \) |
| 67 | \( 1 + 0.919T + 67T^{2} \) |
| 71 | \( 1 - 5.96iT - 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 9.94T + 83T^{2} \) |
| 89 | \( 1 + 13.0iT - 89T^{2} \) |
| 97 | \( 1 - 4.51iT - 97T^{2} \) |
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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
−9.545324606645795308002937096361, −8.411423447121802575619846833193, −7.47647537822065501544263427846, −7.06439104015582775094847143411, −5.83945902098531540241293264948, −4.96753217321047330318836029259, −4.57663249939805580229099885790, −3.20321894167950093825541532545, −2.03362443907821097642971086161, −1.19927154480547723408051761739,
1.82609181731468144935480065808, 2.85149868354634021333788942369, 4.01368283118760656825156582593, 4.73325537370181811441648890308, 5.33843568335217063009402938486, 6.40801586195182160151632285919, 7.31485555546051113394155799038, 7.924838692334409348447467309826, 8.868716310087670303698602051842, 9.989121331797215239173906127695