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 | \( 1 + (-2.40 + 2.40i)T - 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.903707661747016428470564885799, −8.942530841821738751286422820383, −7.86063918929226364012221327236, −7.16125267098066025671933860711, −5.72093146020926619899342365159, −5.29340587454653619054579268147, −4.52060026729801335208984023371, −3.60672063758334083333951872909, −2.04715713997705951745291304921, −0.71033449165130390214359727329,
1.37267965693224011626163588674, 2.88768947930509020856091052607, 4.30986756396570065227155801710, 5.05302177365969718759247028203, 5.66148427645308891101657241342, 6.68877785889503987873320332538, 7.68679100037380103663387610438, 7.83352976340364878302326495231, 9.104460001638564248157247208575, 10.48002126743317659723659821303