L(s) = 1 | + (0.0302 + 0.112i)2-s + 2.59i·3-s + (1.72 − 0.993i)4-s + (0.456 − 1.70i)5-s + (−0.293 + 0.0785i)6-s + (0.329 + 0.329i)8-s − 3.75·9-s + 0.206·10-s + (−1.38 − 1.38i)11-s + (2.58 + 4.47i)12-s + (1.85 + 3.09i)13-s + (4.43 + 1.18i)15-s + (1.95 − 3.39i)16-s + (2.13 + 3.70i)17-s + (−0.113 − 0.423i)18-s + (3.01 + 3.01i)19-s + ⋯ |
L(s) = 1 | + (0.0213 + 0.0797i)2-s + 1.50i·3-s + (0.860 − 0.496i)4-s + (0.204 − 0.762i)5-s + (−0.119 + 0.0320i)6-s + (0.116 + 0.116i)8-s − 1.25·9-s + 0.0651·10-s + (−0.417 − 0.417i)11-s + (0.745 + 1.29i)12-s + (0.515 + 0.857i)13-s + (1.14 + 0.306i)15-s + (0.489 − 0.848i)16-s + (0.518 + 0.898i)17-s + (−0.0267 − 0.0997i)18-s + (0.692 + 0.692i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.71401 + 0.909448i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.71401 + 0.909448i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-1.85 - 3.09i)T \) |
good | 2 | \( 1 + (-0.0302 - 0.112i)T + (-1.73 + i)T^{2} \) |
| 3 | \( 1 - 2.59iT - 3T^{2} \) |
| 5 | \( 1 + (-0.456 + 1.70i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (1.38 + 1.38i)T + 11iT^{2} \) |
| 17 | \( 1 + (-2.13 - 3.70i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.01 - 3.01i)T + 19iT^{2} \) |
| 23 | \( 1 + (-5.53 - 3.19i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.57 + 6.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.13 - 1.10i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-2.73 + 0.732i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.94 + 11.0i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (1.55 + 0.896i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6.40 + 1.71i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (2.13 - 3.70i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.62 + 0.436i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 - 3.08iT - 61T^{2} \) |
| 67 | \( 1 + (-0.0139 + 0.0139i)T - 67iT^{2} \) |
| 71 | \( 1 + (-1.23 - 4.59i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (0.255 + 0.954i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.96 - 5.14i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (9.87 + 9.87i)T + 83iT^{2} \) |
| 89 | \( 1 + (2.07 + 7.76i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (14.2 - 3.82i)T + (84.0 - 48.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.77632692782488700756232445030, −9.794763652915189508253632045956, −9.268086480392125234137295225016, −8.312402834436818169312200835388, −7.13867676341677923601798095474, −5.71783757038694600099828564399, −5.43325515793321250871136409570, −4.19962185851977331236751983488, −3.22121541133903370730522119727, −1.54027957187770361049003990263,
1.25298645073371861075971639485, 2.64765597961651953418134244818, 3.11382631472821032878827262575, 5.19446906916552882821551520247, 6.36774159996397292865133162133, 6.97973614565388930694389608917, 7.55010883347419455457225250107, 8.284235967548683900337029244672, 9.626863972195689409732795298498, 10.87670079693909030295746967097