L(s) = 1 | + (0.707 + 0.707i)2-s + (−1.65 − 0.522i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (−0.797 − 1.53i)6-s + 3.44·7-s + (−0.707 + 0.707i)8-s + (2.45 + 1.72i)9-s − 1.00·10-s − 3.04·11-s + (0.522 − 1.65i)12-s + (0.571 + 0.571i)13-s + (2.43 + 2.43i)14-s + (1.53 − 0.797i)15-s − 1.00·16-s + (1.58 − 1.58i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.953 − 0.301i)3-s + 0.500i·4-s + (−0.316 + 0.316i)5-s + (−0.325 − 0.627i)6-s + 1.30·7-s + (−0.250 + 0.250i)8-s + (0.817 + 0.575i)9-s − 0.316·10-s − 0.918·11-s + (0.150 − 0.476i)12-s + (0.158 + 0.158i)13-s + (0.650 + 0.650i)14-s + (0.396 − 0.205i)15-s − 0.250·16-s + (0.383 − 0.383i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0789 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0789 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.506832884\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.506832884\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.65 + 0.522i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 37 | \( 1 + (1.89 - 5.77i)T \) |
good | 7 | \( 1 - 3.44T + 7T^{2} \) |
| 11 | \( 1 + 3.04T + 11T^{2} \) |
| 13 | \( 1 + (-0.571 - 0.571i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.58 + 1.58i)T - 17iT^{2} \) |
| 19 | \( 1 + (-2.90 - 2.90i)T + 19iT^{2} \) |
| 23 | \( 1 + (-2.05 + 2.05i)T - 23iT^{2} \) |
| 29 | \( 1 + (-1.83 - 1.83i)T + 29iT^{2} \) |
| 31 | \( 1 + (6.33 - 6.33i)T - 31iT^{2} \) |
| 41 | \( 1 - 8.00T + 41T^{2} \) |
| 43 | \( 1 + (-3.17 - 3.17i)T + 43iT^{2} \) |
| 47 | \( 1 - 7.90iT - 47T^{2} \) |
| 53 | \( 1 - 12.4iT - 53T^{2} \) |
| 59 | \( 1 + (2.00 - 2.00i)T - 59iT^{2} \) |
| 61 | \( 1 + (-2.00 + 2.00i)T - 61iT^{2} \) |
| 67 | \( 1 - 9.46iT - 67T^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 4.40iT - 73T^{2} \) |
| 79 | \( 1 + (8.20 + 8.20i)T + 79iT^{2} \) |
| 83 | \( 1 - 10.8iT - 83T^{2} \) |
| 89 | \( 1 + (-9.90 - 9.90i)T + 89iT^{2} \) |
| 97 | \( 1 + (-2.40 - 2.40i)T + 97iT^{2} \) |
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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
−10.48002126743317659723659821303, −9.104460001638564248157247208575, −7.83352976340364878302326495231, −7.68679100037380103663387610438, −6.68877785889503987873320332538, −5.66148427645308891101657241342, −5.05302177365969718759247028203, −4.30986756396570065227155801710, −2.88768947930509020856091052607, −1.37267965693224011626163588674,
0.71033449165130390214359727329, 2.04715713997705951745291304921, 3.60672063758334083333951872909, 4.52060026729801335208984023371, 5.29340587454653619054579268147, 5.72093146020926619899342365159, 7.16125267098066025671933860711, 7.86063918929226364012221327236, 8.942530841821738751286422820383, 9.903707661747016428470564885799