L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.878 − 1.49i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (−1.67 − 0.434i)6-s + 2.38·7-s + (−0.707 − 0.707i)8-s + (−1.45 + 2.62i)9-s + 1.00·10-s + 5.76·11-s + (−1.49 + 0.878i)12-s + (−1.58 + 1.58i)13-s + (1.68 − 1.68i)14-s + (0.434 − 1.67i)15-s − 1.00·16-s + (−0.766 − 0.766i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.507 − 0.861i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (−0.684 − 0.177i)6-s + 0.901·7-s + (−0.250 − 0.250i)8-s + (−0.485 + 0.874i)9-s + 0.316·10-s + 1.73·11-s + (−0.430 + 0.253i)12-s + (−0.438 + 0.438i)13-s + (0.450 − 0.450i)14-s + (0.112 − 0.432i)15-s − 0.250·16-s + (−0.186 − 0.186i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.251244665\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.251244665\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.878 + 1.49i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 37 | \( 1 + (-5.79 + 1.85i)T \) |
good | 7 | \( 1 - 2.38T + 7T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 13 | \( 1 + (1.58 - 1.58i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.766 + 0.766i)T + 17iT^{2} \) |
| 19 | \( 1 + (-4.27 + 4.27i)T - 19iT^{2} \) |
| 23 | \( 1 + (-5.87 - 5.87i)T + 23iT^{2} \) |
| 29 | \( 1 + (0.666 - 0.666i)T - 29iT^{2} \) |
| 31 | \( 1 + (4.93 + 4.93i)T + 31iT^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 + (6.47 - 6.47i)T - 43iT^{2} \) |
| 47 | \( 1 + 8.44iT - 47T^{2} \) |
| 53 | \( 1 - 7.21iT - 53T^{2} \) |
| 59 | \( 1 + (5.63 + 5.63i)T + 59iT^{2} \) |
| 61 | \( 1 + (8.82 + 8.82i)T + 61iT^{2} \) |
| 67 | \( 1 + 2.96iT - 67T^{2} \) |
| 71 | \( 1 + 10.2iT - 71T^{2} \) |
| 73 | \( 1 + 1.48iT - 73T^{2} \) |
| 79 | \( 1 + (3.85 - 3.85i)T - 79iT^{2} \) |
| 83 | \( 1 - 16.2iT - 83T^{2} \) |
| 89 | \( 1 + (3.88 - 3.88i)T - 89iT^{2} \) |
| 97 | \( 1 + (-11.4 + 11.4i)T - 97iT^{2} \) |
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
−9.509855299766313439394795011562, −9.146090770572154553658984430013, −7.72776517247304260177331057013, −7.02191565970408603800892065228, −6.29545758938073367661481643180, −5.30009888549124847938070022903, −4.57587386504736256664593042321, −3.25574239566084489713644307416, −1.97927980149927188212505988401, −1.16045647555273202155420515377,
1.32370477216399643352582182071, 3.15484065276119191299694546095, 4.20617949463455724247839597277, 4.84555793291087919636442161778, 5.66493533545734920544488519840, 6.44846572859651642943403263278, 7.40168632560391939103111554401, 8.572938135818179945124100224344, 9.099758173060531467441825255019, 9.994080657110618551293996763987