L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.0444 + 1.73i)3-s + 1.00i·4-s + (−0.142 + 2.23i)5-s + (1.19 − 1.25i)6-s + (2.34 − 2.34i)7-s + (0.707 − 0.707i)8-s + (−2.99 + 0.153i)9-s + (1.67 − 1.47i)10-s − 5.32i·11-s + (−1.73 + 0.0444i)12-s + (−2.57 − 2.57i)13-s − 3.31·14-s + (−3.87 − 0.147i)15-s − 1.00·16-s + (−2.33 − 2.33i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.0256 + 0.999i)3-s + 0.500i·4-s + (−0.0635 + 0.997i)5-s + (0.487 − 0.512i)6-s + (0.885 − 0.885i)7-s + (0.250 − 0.250i)8-s + (−0.998 + 0.0512i)9-s + (0.530 − 0.467i)10-s − 1.60i·11-s + (−0.499 + 0.0128i)12-s + (−0.714 − 0.714i)13-s − 0.885·14-s + (−0.999 − 0.0379i)15-s − 0.250·16-s + (−0.565 − 0.565i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.666944 - 0.548748i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.666944 - 0.548748i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.0444 - 1.73i)T \) |
| 5 | \( 1 + (0.142 - 2.23i)T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + (-2.34 + 2.34i)T - 7iT^{2} \) |
| 11 | \( 1 + 5.32iT - 11T^{2} \) |
| 13 | \( 1 + (2.57 + 2.57i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.33 + 2.33i)T + 17iT^{2} \) |
| 19 | \( 1 + 5.87iT - 19T^{2} \) |
| 23 | \( 1 + (-2.05 + 2.05i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 37 | \( 1 + (-0.718 + 0.718i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.62iT - 41T^{2} \) |
| 43 | \( 1 + (-1.71 - 1.71i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.49 + 6.49i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.55 + 5.55i)T - 53iT^{2} \) |
| 59 | \( 1 - 6.86T + 59T^{2} \) |
| 61 | \( 1 + 0.913T + 61T^{2} \) |
| 67 | \( 1 + (9.71 - 9.71i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.59iT - 71T^{2} \) |
| 73 | \( 1 + (5.80 + 5.80i)T + 73iT^{2} \) |
| 79 | \( 1 - 10.0iT - 79T^{2} \) |
| 83 | \( 1 + (-4.35 + 4.35i)T - 83iT^{2} \) |
| 89 | \( 1 - 3.15T + 89T^{2} \) |
| 97 | \( 1 + (7.10 - 7.10i)T - 97iT^{2} \) |
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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
−10.12447844397876227318684373601, −9.170136495587610795883591929411, −8.338362711074078902674193522286, −7.56838422146691020852174952956, −6.58372231429355862623439568100, −5.30418748035532360283616898432, −4.37108127465346617225558148942, −3.29627476060815956550186300815, −2.60989885120156006503097326810, −0.46929109999543346915906715213,
1.64208110670971623071334081872, 2.07077733536812695970712775925, 4.31657696952969189312722096611, 5.21753453327892784955900722494, 5.95625574030827983302283842052, 7.14955224039708257077611328897, 7.69819077355017070220699190266, 8.533906324551665294149350918842, 9.113692293978879419897039438235, 9.945165758027632636326943975993