| L(s) = 1 | + 2·5-s + 12·7-s + 60·11-s + 42·13-s − 10·17-s − 132·19-s + 48·23-s − 121·25-s + 226·29-s − 252·31-s + 24·35-s + 362·37-s + 94·41-s + 228·43-s + 408·47-s − 199·49-s + 346·53-s + 120·55-s − 300·59-s + 466·61-s + 84·65-s − 204·67-s − 1.05e3·71-s + 330·73-s + 720·77-s + 612·79-s + 564·83-s + ⋯ |
| L(s) = 1 | + 0.178·5-s + 0.647·7-s + 1.64·11-s + 0.896·13-s − 0.142·17-s − 1.59·19-s + 0.435·23-s − 0.967·25-s + 1.44·29-s − 1.46·31-s + 0.115·35-s + 1.60·37-s + 0.358·41-s + 0.808·43-s + 1.26·47-s − 0.580·49-s + 0.896·53-s + 0.294·55-s − 0.661·59-s + 0.978·61-s + 0.160·65-s − 0.371·67-s − 1.76·71-s + 0.529·73-s + 1.06·77-s + 0.871·79-s + 0.745·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.518584386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.518584386\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 + 10 T + p^{3} T^{2} \) |
| 19 | \( 1 + 132 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 226 T + p^{3} T^{2} \) |
| 31 | \( 1 + 252 T + p^{3} T^{2} \) |
| 37 | \( 1 - 362 T + p^{3} T^{2} \) |
| 41 | \( 1 - 94 T + p^{3} T^{2} \) |
| 43 | \( 1 - 228 T + p^{3} T^{2} \) |
| 47 | \( 1 - 408 T + p^{3} T^{2} \) |
| 53 | \( 1 - 346 T + p^{3} T^{2} \) |
| 59 | \( 1 + 300 T + p^{3} T^{2} \) |
| 61 | \( 1 - 466 T + p^{3} T^{2} \) |
| 67 | \( 1 + 204 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1056 T + p^{3} T^{2} \) |
| 73 | \( 1 - 330 T + p^{3} T^{2} \) |
| 79 | \( 1 - 612 T + p^{3} T^{2} \) |
| 83 | \( 1 - 564 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1510 T + p^{3} T^{2} \) |
| 97 | \( 1 - 594 T + p^{3} T^{2} \) |
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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
−10.45843824000408707268853035107, −9.210189198507934451695928357481, −8.739525932929834879856884027317, −7.70119578558712549437530055538, −6.53418907636563243044812481763, −5.93752925222959220047766767340, −4.49511438857260292053011677693, −3.79754085113831870496611223449, −2.15564171723596648911957826527, −1.02949376167775775981670093000,
1.02949376167775775981670093000, 2.15564171723596648911957826527, 3.79754085113831870496611223449, 4.49511438857260292053011677693, 5.93752925222959220047766767340, 6.53418907636563243044812481763, 7.70119578558712549437530055538, 8.739525932929834879856884027317, 9.210189198507934451695928357481, 10.45843824000408707268853035107
