| L(s) = 1 | + 18i·7-s + 34·11-s + 12i·13-s + 102i·17-s − 164·19-s + 48i·23-s − 146·29-s + 100·31-s − 328i·37-s − 288·41-s + 120i·43-s − 16i·47-s + 19·49-s − 126i·53-s − 642·59-s + ⋯ |
| L(s) = 1 | + 0.971i·7-s + 0.931·11-s + 0.256i·13-s + 1.45i·17-s − 1.98·19-s + 0.435i·23-s − 0.934·29-s + 0.579·31-s − 1.45i·37-s − 1.09·41-s + 0.425i·43-s − 0.0496i·47-s + 0.0553·49-s − 0.326i·53-s − 1.41·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3153013778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3153013778\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 18iT - 343T^{2} \) |
| 11 | \( 1 - 34T + 1.33e3T^{2} \) |
| 13 | \( 1 - 12iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 102iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 164T + 6.85e3T^{2} \) |
| 23 | \( 1 - 48iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 146T + 2.43e4T^{2} \) |
| 31 | \( 1 - 100T + 2.97e4T^{2} \) |
| 37 | \( 1 + 328iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 288T + 6.89e4T^{2} \) |
| 43 | \( 1 - 120iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 16iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 126iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 642T + 2.05e5T^{2} \) |
| 61 | \( 1 - 602T + 2.26e5T^{2} \) |
| 67 | \( 1 + 436iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 652T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.06e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 388T + 4.93e5T^{2} \) |
| 83 | \( 1 + 444iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 820T + 7.04e5T^{2} \) |
| 97 | \( 1 - 766iT - 9.12e5T^{2} \) |
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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
−9.168876540196448358245094479890, −8.669003200052254935327035754423, −7.991687234864197156832767677674, −6.78107458805227617259886829369, −6.20800618494999811490327258087, −5.48090354492161433327935771825, −4.28684435985304651758825766701, −3.66506049336864269401826874791, −2.29202952162701447076323758834, −1.60770701664398782225133602978,
0.06784971520862014483734632499, 1.10468727726316484912203662038, 2.31219757503544581686062537829, 3.48635298076517765561621587986, 4.30374673589243183873231245239, 5.01982741804306520870234579726, 6.31936753128841503230634265005, 6.78534477601093108375514833334, 7.62460883813928955829211532162, 8.504583306019125323992786528694
