L(s) = 1 | + (−1 − i)2-s + (−1.58 − 1.58i)3-s + (1.58 + 1.58i)5-s + 3.16i·6-s + (2.58 − 0.581i)7-s + (−2 + 2i)8-s + 2.00i·9-s − 3.16i·10-s − 11-s + (1.58 + 1.58i)13-s + (−3.16 − 2i)14-s − 5.00i·15-s + 4·16-s + (−1.58 + 1.58i)17-s + (2.00 − 2.00i)18-s − 3.16·19-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.912 − 0.912i)3-s + (0.707 + 0.707i)5-s + 1.29i·6-s + (0.975 − 0.219i)7-s + (−0.707 + 0.707i)8-s + 0.666i·9-s − 1.00i·10-s − 0.301·11-s + (0.438 + 0.438i)13-s + (−0.845 − 0.534i)14-s − 1.29i·15-s + 16-s + (−0.383 + 0.383i)17-s + (0.471 − 0.471i)18-s − 0.725·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0103 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0103 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.342239 - 0.345806i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.342239 - 0.345806i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.58 - 1.58i)T \) |
| 7 | \( 1 + (-2.58 + 0.581i)T \) |
good | 2 | \( 1 + (1 + i)T + 2iT^{2} \) |
| 3 | \( 1 + (1.58 + 1.58i)T + 3iT^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + (-1.58 - 1.58i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.58 - 1.58i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.16T + 19T^{2} \) |
| 23 | \( 1 + (-2 + 2i)T - 23iT^{2} \) |
| 29 | \( 1 - 3iT - 29T^{2} \) |
| 31 | \( 1 - 3.16iT - 31T^{2} \) |
| 37 | \( 1 + (6 + 6i)T + 37iT^{2} \) |
| 41 | \( 1 + 9.48iT - 41T^{2} \) |
| 43 | \( 1 + (3 - 3i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.74 + 4.74i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1 + i)T - 53iT^{2} \) |
| 59 | \( 1 + 9.48T + 59T^{2} \) |
| 61 | \( 1 - 6.32iT - 61T^{2} \) |
| 67 | \( 1 + (1 + i)T + 67iT^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 73iT^{2} \) |
| 79 | \( 1 - 13iT - 79T^{2} \) |
| 83 | \( 1 + (-3.16 - 3.16i)T + 83iT^{2} \) |
| 89 | \( 1 - 6.32T + 89T^{2} \) |
| 97 | \( 1 + (-1.58 + 1.58i)T - 97iT^{2} \) |
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
−17.07249777995446349307034207198, −14.95842382195979093515524107496, −13.81862293297963892515678402907, −12.32676045744168026200468324578, −11.06694238668489043531651720064, −10.58209303738455575164574949203, −8.784546719314304664614993136090, −6.93564282090983517655828207269, −5.61138339060652986428905421835, −1.81797194049574087688155293048,
4.76830288810563033065637744208, 6.04415488762146579249464185117, 8.071770588726477605787112555276, 9.238250136225366813809977809960, 10.50833723239882525662713597838, 11.83616758473911985107263360753, 13.31589695561115667447681758897, 15.16786410824094682503509894907, 16.00897177664703438570236490536, 17.09459915768791383421083954730