| L(s) = 1 | + 3·3-s + 18·5-s − 7·7-s + 9·9-s + 72·11-s − 34·13-s + 54·15-s + 6·17-s − 92·19-s − 21·21-s + 180·23-s + 199·25-s + 27·27-s − 114·29-s − 56·31-s + 216·33-s − 126·35-s − 34·37-s − 102·39-s + 6·41-s − 164·43-s + 162·45-s − 168·47-s + 49·49-s + 18·51-s + 654·53-s + 1.29e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.60·5-s − 0.377·7-s + 1/3·9-s + 1.97·11-s − 0.725·13-s + 0.929·15-s + 0.0856·17-s − 1.11·19-s − 0.218·21-s + 1.63·23-s + 1.59·25-s + 0.192·27-s − 0.729·29-s − 0.324·31-s + 1.13·33-s − 0.608·35-s − 0.151·37-s − 0.418·39-s + 0.0228·41-s − 0.581·43-s + 0.536·45-s − 0.521·47-s + 1/7·49-s + 0.0494·51-s + 1.69·53-s + 3.17·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.167690738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.167690738\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 72 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 6 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 180 T + p^{3} T^{2} \) |
| 29 | \( 1 + 114 T + p^{3} T^{2} \) |
| 31 | \( 1 + 56 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 168 T + p^{3} T^{2} \) |
| 53 | \( 1 - 654 T + p^{3} T^{2} \) |
| 59 | \( 1 - 492 T + p^{3} T^{2} \) |
| 61 | \( 1 + 250 T + p^{3} T^{2} \) |
| 67 | \( 1 - 124 T + p^{3} T^{2} \) |
| 71 | \( 1 + 36 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1010 T + p^{3} T^{2} \) |
| 79 | \( 1 + 56 T + p^{3} T^{2} \) |
| 83 | \( 1 + 228 T + p^{3} T^{2} \) |
| 89 | \( 1 - 390 T + p^{3} T^{2} \) |
| 97 | \( 1 + 70 T + p^{3} T^{2} \) |
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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
−10.97063371379144635756312674025, −9.887411038356644898325913128793, −9.309185910214281596244060836286, −8.704711987937490937472325150768, −6.98841003189115270424790114680, −6.44031836747561565089771180367, −5.23966758715370911869091302667, −3.85577483848919244336909269014, −2.46611696453234412923932194191, −1.37817848240816738073129413396,
1.37817848240816738073129413396, 2.46611696453234412923932194191, 3.85577483848919244336909269014, 5.23966758715370911869091302667, 6.44031836747561565089771180367, 6.98841003189115270424790114680, 8.704711987937490937472325150768, 9.309185910214281596244060836286, 9.887411038356644898325913128793, 10.97063371379144635756312674025
