| L(s) = 1 | + 9·5-s − 7-s + 63·11-s − 28·13-s + 72·17-s + 98·19-s + 126·23-s − 44·25-s − 126·29-s − 259·31-s − 9·35-s + 386·37-s − 450·41-s − 34·43-s − 54·47-s − 342·49-s − 693·53-s + 567·55-s + 180·59-s − 280·61-s − 252·65-s − 586·67-s + 504·71-s + 161·73-s − 63·77-s + 440·79-s + 999·83-s + ⋯ |
| L(s) = 1 | + 0.804·5-s − 0.0539·7-s + 1.72·11-s − 0.597·13-s + 1.02·17-s + 1.18·19-s + 1.14·23-s − 0.351·25-s − 0.806·29-s − 1.50·31-s − 0.0434·35-s + 1.71·37-s − 1.71·41-s − 0.120·43-s − 0.167·47-s − 0.997·49-s − 1.79·53-s + 1.39·55-s + 0.397·59-s − 0.587·61-s − 0.480·65-s − 1.06·67-s + 0.842·71-s + 0.258·73-s − 0.0932·77-s + 0.626·79-s + 1.32·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.900432964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.900432964\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 9 T + p^{3} T^{2} \) |
| 7 | \( 1 + T + p^{3} T^{2} \) |
| 11 | \( 1 - 63 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 72 T + p^{3} T^{2} \) |
| 19 | \( 1 - 98 T + p^{3} T^{2} \) |
| 23 | \( 1 - 126 T + p^{3} T^{2} \) |
| 29 | \( 1 + 126 T + p^{3} T^{2} \) |
| 31 | \( 1 + 259 T + p^{3} T^{2} \) |
| 37 | \( 1 - 386 T + p^{3} T^{2} \) |
| 41 | \( 1 + 450 T + p^{3} T^{2} \) |
| 43 | \( 1 + 34 T + p^{3} T^{2} \) |
| 47 | \( 1 + 54 T + p^{3} T^{2} \) |
| 53 | \( 1 + 693 T + p^{3} T^{2} \) |
| 59 | \( 1 - 180 T + p^{3} T^{2} \) |
| 61 | \( 1 + 280 T + p^{3} T^{2} \) |
| 67 | \( 1 + 586 T + p^{3} T^{2} \) |
| 71 | \( 1 - 504 T + p^{3} T^{2} \) |
| 73 | \( 1 - 161 T + p^{3} T^{2} \) |
| 79 | \( 1 - 440 T + p^{3} T^{2} \) |
| 83 | \( 1 - 999 T + p^{3} T^{2} \) |
| 89 | \( 1 - 882 T + p^{3} T^{2} \) |
| 97 | \( 1 + 721 T + p^{3} T^{2} \) |
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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
−13.28917559913371105151489613547, −12.15288330283312116431885897508, −11.22439837725103309426666104910, −9.685695511502878639308345925707, −9.290702780966852884685169905925, −7.58802248976385018640100288420, −6.39057132125290198952260909842, −5.18406504272717641399117477668, −3.44307730524507891492569170438, −1.48366022759503054930530043772,
1.48366022759503054930530043772, 3.44307730524507891492569170438, 5.18406504272717641399117477668, 6.39057132125290198952260909842, 7.58802248976385018640100288420, 9.290702780966852884685169905925, 9.685695511502878639308345925707, 11.22439837725103309426666104910, 12.15288330283312116431885897508, 13.28917559913371105151489613547
