L(s) = 1 | − 2-s + (−1.65 − 0.509i)3-s + 4-s + 2.90i·5-s + (1.65 + 0.509i)6-s − 3.06·7-s − 8-s + (2.48 + 1.68i)9-s − 2.90i·10-s + 1.35·11-s + (−1.65 − 0.509i)12-s − 7.06i·13-s + 3.06·14-s + (1.48 − 4.80i)15-s + 16-s + 0.568i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.955 − 0.294i)3-s + 0.5·4-s + 1.29i·5-s + (0.675 + 0.208i)6-s − 1.16·7-s − 0.353·8-s + (0.826 + 0.562i)9-s − 0.917i·10-s + 0.408·11-s + (−0.477 − 0.147i)12-s − 1.95i·13-s + 0.820·14-s + (0.382 − 1.24i)15-s + 0.250·16-s + 0.137i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0771695 - 0.168225i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0771695 - 0.168225i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (1.65 + 0.509i)T \) |
| 59 | \( 1 + (3.07 - 7.03i)T \) |
good | 5 | \( 1 - 2.90iT - 5T^{2} \) |
| 7 | \( 1 + 3.06T + 7T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 13 | \( 1 + 7.06iT - 13T^{2} \) |
| 17 | \( 1 - 0.568iT - 17T^{2} \) |
| 19 | \( 1 + 4.42T + 19T^{2} \) |
| 23 | \( 1 + 7.20T + 23T^{2} \) |
| 29 | \( 1 + 2.41iT - 29T^{2} \) |
| 31 | \( 1 + 6.44iT - 31T^{2} \) |
| 37 | \( 1 + 1.74iT - 37T^{2} \) |
| 41 | \( 1 + 8.09iT - 41T^{2} \) |
| 43 | \( 1 + 0.770iT - 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 + 1.03iT - 53T^{2} \) |
| 61 | \( 1 - 10.6iT - 61T^{2} \) |
| 67 | \( 1 + 1.09iT - 67T^{2} \) |
| 71 | \( 1 + 11.0iT - 71T^{2} \) |
| 73 | \( 1 - 5.96iT - 73T^{2} \) |
| 79 | \( 1 - 8.62T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 8.16iT - 97T^{2} \) |
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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
−10.77949573523982990776007732215, −10.39800936302582037568586425769, −9.679437953916926383254515249284, −8.095382421159393350405615997642, −7.23902417612831251061127717250, −6.29519409275639894133831699625, −5.84088148432443818301566409165, −3.75261485573593655758622436758, −2.45531379445463472890451824723, −0.17678319186235787895191156627,
1.58219246825616697157073117853, 3.90927441675529685744337903036, 4.87720500921841065151384919377, 6.33070967848373999932579008822, 6.70378028772140510325561062858, 8.332230590295464417747235007464, 9.361199892468444654991654407063, 9.639367095281897060055677130481, 10.83749523356389602737419275308, 11.89476800439326094058070956356