L(s) = 1 | + (1.47 − 1.47i)3-s + (55.2 − 8.52i)5-s + (156. + 156. i)7-s + 238. i·9-s + 35.4i·11-s + (−247. − 247. i)13-s + (69.0 − 94.2i)15-s + (−1.19e3 + 1.19e3i)17-s − 1.01e3·19-s + 461.·21-s + (2.03e3 − 2.03e3i)23-s + (2.97e3 − 942. i)25-s + (711. + 711. i)27-s + 2.20e3i·29-s + 6.17e3i·31-s + ⋯ |
L(s) = 1 | + (0.0947 − 0.0947i)3-s + (0.988 − 0.152i)5-s + (1.20 + 1.20i)7-s + 0.982i·9-s + 0.0884i·11-s + (−0.405 − 0.405i)13-s + (0.0792 − 0.108i)15-s + (−1.00 + 1.00i)17-s − 0.642·19-s + 0.228·21-s + (0.801 − 0.801i)23-s + (0.953 − 0.301i)25-s + (0.187 + 0.187i)27-s + 0.487i·29-s + 1.15i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.492505446\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.492505446\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-55.2 + 8.52i)T \) |
good | 3 | \( 1 + (-1.47 + 1.47i)T - 243iT^{2} \) |
| 7 | \( 1 + (-156. - 156. i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 - 35.4iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (247. + 247. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (1.19e3 - 1.19e3i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 + 1.01e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-2.03e3 + 2.03e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 - 2.20e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 6.17e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (9.46e3 - 9.46e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 + 9.00e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.59e4 + 1.59e4i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (-7.19e3 - 7.19e3i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (1.29e4 + 1.29e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 - 4.05e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.92e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-1.82e4 - 1.82e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 2.66e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-3.64e4 - 3.64e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 + 5.08e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-2.03e4 + 2.03e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 5.93e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-6.95e4 + 6.95e4i)T - 8.58e9iT^{2} \) |
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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
−12.33558258175009710750465817305, −11.00323219282556043062918914572, −10.32982970653936797272857069406, −8.730621887931952302780800602294, −8.438486334148420902779161275909, −6.80146407523161995586810302047, −5.44667599984356612209801126057, −4.80672884243087302335856843164, −2.46320163188241689481934650008, −1.71961416883680084948431854474,
0.819059916348767244595269756700, 2.21796798113967994700151704038, 3.99562826290536771647064027847, 5.11147925617091287673683831735, 6.56961915285117737989583067064, 7.44376874399030134724102309059, 8.901732547901923429064955022673, 9.719867152481433492651542434124, 10.84488493134352286905921037450, 11.56543955315306713484195001746