L(s) = 1 | + 1.90i·2-s + (−1.14 − 1.14i)3-s − 1.62·4-s + (−2.29 − 2.29i)5-s + (2.17 − 2.17i)6-s + (3.32 − 3.32i)7-s + 0.709i·8-s − 0.399i·9-s + (4.36 − 4.36i)10-s + (−0.707 + 0.707i)11-s + (1.85 + 1.85i)12-s + 2.85·13-s + (6.34 + 6.34i)14-s + 5.22i·15-s − 4.60·16-s + (−3.62 − 1.96i)17-s + ⋯ |
L(s) = 1 | + 1.34i·2-s + (−0.658 − 0.658i)3-s − 0.813·4-s + (−1.02 − 1.02i)5-s + (0.886 − 0.886i)6-s + (1.25 − 1.25i)7-s + 0.250i·8-s − 0.133i·9-s + (1.38 − 1.38i)10-s + (−0.213 + 0.213i)11-s + (0.535 + 0.535i)12-s + 0.791·13-s + (1.69 + 1.69i)14-s + 1.34i·15-s − 1.15·16-s + (−0.878 − 0.476i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.841177 - 0.159471i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.841177 - 0.159471i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.707 - 0.707i)T \) |
| 17 | \( 1 + (3.62 + 1.96i)T \) |
good | 2 | \( 1 - 1.90iT - 2T^{2} \) |
| 3 | \( 1 + (1.14 + 1.14i)T + 3iT^{2} \) |
| 5 | \( 1 + (2.29 + 2.29i)T + 5iT^{2} \) |
| 7 | \( 1 + (-3.32 + 3.32i)T - 7iT^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 19 | \( 1 + 3.83iT - 19T^{2} \) |
| 23 | \( 1 + (-3.28 + 3.28i)T - 23iT^{2} \) |
| 29 | \( 1 + (-3.14 - 3.14i)T + 29iT^{2} \) |
| 31 | \( 1 + (-1.43 - 1.43i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.52 - 1.52i)T + 37iT^{2} \) |
| 41 | \( 1 + (-5.52 + 5.52i)T - 41iT^{2} \) |
| 43 | \( 1 - 12.7iT - 43T^{2} \) |
| 47 | \( 1 + 5.54T + 47T^{2} \) |
| 53 | \( 1 - 4.51iT - 53T^{2} \) |
| 59 | \( 1 - 0.104iT - 59T^{2} \) |
| 61 | \( 1 + (-3.14 + 3.14i)T - 61iT^{2} \) |
| 67 | \( 1 + 2.20T + 67T^{2} \) |
| 71 | \( 1 + (-3.31 - 3.31i)T + 71iT^{2} \) |
| 73 | \( 1 + (-2.33 - 2.33i)T + 73iT^{2} \) |
| 79 | \( 1 + (4.15 - 4.15i)T - 79iT^{2} \) |
| 83 | \( 1 - 15.1iT - 83T^{2} \) |
| 89 | \( 1 - 18.3T + 89T^{2} \) |
| 97 | \( 1 + (3.71 + 3.71i)T + 97iT^{2} \) |
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
−12.67962667364860509911894778076, −11.40316184235823303519994518345, −11.07047560233223341080267614294, −8.903868232254994056006343326709, −8.119847339404515475871704310209, −7.30966873371866068017172083006, −6.55550078996858972059114017292, −4.99925610626819659635969018992, −4.40583461612377421874355772396, −0.896547421737418930582258290921,
2.18222217030183084337722211746, 3.60275575964924863083575997768, 4.71926650331543926002509011708, 6.09390524996683481586456543529, 7.77979194365888000395148415578, 8.819729750496429056804650574136, 10.28667661717074202989463151018, 11.00215557333200901811107516690, 11.45586959587126702186438331327, 11.96831149342838338624745276142