L(s) = 1 | + 2.97i·3-s + (2.21 + 0.282i)5-s − 4.30i·7-s − 5.87·9-s − 1.96·11-s − 5.83i·13-s + (−0.842 + 6.60i)15-s − 6.46i·17-s + 19-s + 12.8·21-s + 4.45i·23-s + (4.84 + 1.25i)25-s − 8.55i·27-s + 5.31·29-s + 0.713·31-s + ⋯ |
L(s) = 1 | + 1.71i·3-s + (0.991 + 0.126i)5-s − 1.62i·7-s − 1.95·9-s − 0.591·11-s − 1.61i·13-s + (−0.217 + 1.70i)15-s − 1.56i·17-s + 0.229·19-s + 2.79·21-s + 0.929i·23-s + (0.968 + 0.250i)25-s − 1.64i·27-s + 0.986·29-s + 0.128·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.726821481\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.726821481\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.21 - 0.282i)T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2.97iT - 3T^{2} \) |
| 7 | \( 1 + 4.30iT - 7T^{2} \) |
| 11 | \( 1 + 1.96T + 11T^{2} \) |
| 13 | \( 1 + 5.83iT - 13T^{2} \) |
| 17 | \( 1 + 6.46iT - 17T^{2} \) |
| 23 | \( 1 - 4.45iT - 23T^{2} \) |
| 29 | \( 1 - 5.31T + 29T^{2} \) |
| 31 | \( 1 - 0.713T + 31T^{2} \) |
| 37 | \( 1 + 8.56iT - 37T^{2} \) |
| 41 | \( 1 - 2.71T + 41T^{2} \) |
| 43 | \( 1 + 7.62iT - 43T^{2} \) |
| 47 | \( 1 - 4.25iT - 47T^{2} \) |
| 53 | \( 1 - 7.90iT - 53T^{2} \) |
| 59 | \( 1 + 6.04T + 59T^{2} \) |
| 61 | \( 1 - 1.00T + 61T^{2} \) |
| 67 | \( 1 + 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 0.272T + 71T^{2} \) |
| 73 | \( 1 - 15.0iT - 73T^{2} \) |
| 79 | \( 1 - 1.95T + 79T^{2} \) |
| 83 | \( 1 + 5.52iT - 83T^{2} \) |
| 89 | \( 1 - 0.632T + 89T^{2} \) |
| 97 | \( 1 + 7.10iT - 97T^{2} \) |
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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
−9.719590370943802344105084393262, −9.015789021437394386115496441407, −7.83763443911215802522831926054, −7.14776210684695985028981564635, −5.80550536597907901079283188122, −5.20165000001728733776088709525, −4.49425320961301795221064277308, −3.42695335812029634632246483011, −2.77067150806109040614620983905, −0.68653900862786583924972143559,
1.47998542322941360550245029470, 2.13959376575351553509306221417, 2.83075272585339205765970438765, 4.74271749010164492370950361305, 5.73779464935081173618934899384, 6.37076663191431176133294549075, 6.70964562686859445403117197933, 8.091831782999238585827085881717, 8.523257720014758451843618771825, 9.203344701411440578773909683238