| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 11-s − 14-s + 16-s − 17-s − 19-s + 20-s + 22-s − 23-s + 25-s − 28-s − 29-s − 31-s + 32-s − 34-s − 35-s − 37-s − 38-s + 40-s + 41-s + 43-s + 44-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 11-s − 14-s + 16-s − 17-s − 19-s + 20-s + 22-s − 23-s + 25-s − 28-s − 29-s − 31-s + 32-s − 34-s − 35-s − 37-s − 38-s + 40-s + 41-s + 43-s + 44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.776581434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.776581434\) |
| \(L(1)\) |
\(\approx\) |
\(2.012229726\) |
| \(L(1)\) |
\(\approx\) |
\(2.012229726\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−34.7455728898393599410470802854, −33.36575152698929719192568831969, −32.654851397097391171163639940629, −31.63378022587838460339783881324, −30.071211749866349843068044079398, −29.38995764131271612067805504454, −28.21670238714452437561428321609, −26.080358633423214338465013260801, −25.2354638079280302768014451571, −24.092599756848773483522062032142, −22.47568677187093845556963849738, −21.93315191768140742388488531843, −20.48894258051107947462916786866, −19.28509033192190916774412727849, −17.337565702284410625359178605047, −16.14464333650454194819340125922, −14.63812553931791531619782220267, −13.467960588076771502489339540526, −12.46016779683215014834642518471, −10.75293335203503761289374520253, −9.26505664657555325046029148131, −6.79945605063326105935862116884, −5.85202180427453258469290681861, −3.955373319394198948596812494220, −2.15367543022529411299025556090,
2.15367543022529411299025556090, 3.955373319394198948596812494220, 5.85202180427453258469290681861, 6.79945605063326105935862116884, 9.26505664657555325046029148131, 10.75293335203503761289374520253, 12.46016779683215014834642518471, 13.467960588076771502489339540526, 14.63812553931791531619782220267, 16.14464333650454194819340125922, 17.337565702284410625359178605047, 19.28509033192190916774412727849, 20.48894258051107947462916786866, 21.93315191768140742388488531843, 22.47568677187093845556963849738, 24.092599756848773483522062032142, 25.2354638079280302768014451571, 26.080358633423214338465013260801, 28.21670238714452437561428321609, 29.38995764131271612067805504454, 30.071211749866349843068044079398, 31.63378022587838460339783881324, 32.654851397097391171163639940629, 33.36575152698929719192568831969, 34.7455728898393599410470802854
