L(s) = 1 | + (4.58 + 4.58i)2-s + (0.411 − 0.411i)3-s − 21.8i·4-s + (−123. − 17.0i)5-s + 3.77·6-s + (215. + 215. i)7-s + (394. − 394. i)8-s + 728. i·9-s + (−489. − 646. i)10-s − 1.32e3·11-s + (−9.00 − 9.00i)12-s + (1.43e3 − 1.43e3i)13-s + 1.97e3i·14-s + (−57.9 + 43.9i)15-s + 2.21e3·16-s + (1.34e3 + 1.34e3i)17-s + ⋯ |
L(s) = 1 | + (0.573 + 0.573i)2-s + (0.0152 − 0.0152i)3-s − 0.341i·4-s + (−0.990 − 0.136i)5-s + 0.0174·6-s + (0.628 + 0.628i)7-s + (0.769 − 0.769i)8-s + 0.999i·9-s + (−0.489 − 0.646i)10-s − 0.996·11-s + (−0.00520 − 0.00520i)12-s + (0.655 − 0.655i)13-s + 0.720i·14-s + (−0.0171 + 0.0130i)15-s + 0.541·16-s + (0.274 + 0.274i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.24028 + 0.262216i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24028 + 0.262216i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (123. + 17.0i)T \) |
good | 2 | \( 1 + (-4.58 - 4.58i)T + 64iT^{2} \) |
| 3 | \( 1 + (-0.411 + 0.411i)T - 729iT^{2} \) |
| 7 | \( 1 + (-215. - 215. i)T + 1.17e5iT^{2} \) |
| 11 | \( 1 + 1.32e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + (-1.43e3 + 1.43e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + (-1.34e3 - 1.34e3i)T + 2.41e7iT^{2} \) |
| 19 | \( 1 + 9.19e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + (7.49e3 - 7.49e3i)T - 1.48e8iT^{2} \) |
| 29 | \( 1 - 5.43e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 3.27e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + (-5.65e4 - 5.65e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 - 1.77e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + (6.39e3 - 6.39e3i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + (3.52e4 + 3.52e4i)T + 1.07e10iT^{2} \) |
| 53 | \( 1 + (6.04e4 - 6.04e4i)T - 2.21e10iT^{2} \) |
| 59 | \( 1 + 4.60e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 7.56e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + (-1.54e5 - 1.54e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 1.84e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-3.01e5 + 3.01e5i)T - 1.51e11iT^{2} \) |
| 79 | \( 1 - 1.78e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (-5.65e5 + 5.65e5i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 - 2.49e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-3.06e4 - 3.06e4i)T + 8.32e11iT^{2} \) |
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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
−23.38506031867736604776694607259, −21.91988837727605791418011938094, −19.91801904767663239097347286980, −18.50359540752517275670456818036, −16.05188646408227432583105387903, −15.09544293120487037724998384683, −13.23573897879801250106247432929, −10.96914804494141550496753799704, −7.897656243489242424922631500684, −5.12914450680140621565638995670,
3.93006190631750773769310300640, 7.88298676318602125850158821797, 11.07532374756711977603541735835, 12.45522604068505471104036455987, 14.36733487052311220638803175645, 16.34116157394896496198909899820, 18.30517695163686365419746645865, 20.31252717827085225860310457320, 21.09781237482003825936515479258, 23.01672011611124721621304828340