| L(s) = 1 | − 0.364i·3-s + 2.86·9-s + 1.36·11-s − 2.63i·13-s − 2.23i·17-s + 6.23·19-s + 6.59i·23-s − 2.13i·27-s − 5.50·29-s + 4.50·31-s − 0.497i·33-s + 2.09i·37-s − 0.960·39-s + 9.32·41-s − 1.86i·43-s + ⋯ |
| L(s) = 1 | − 0.210i·3-s + 0.955·9-s + 0.411·11-s − 0.730i·13-s − 0.541i·17-s + 1.42·19-s + 1.37i·23-s − 0.411i·27-s − 1.02·29-s + 0.808·31-s − 0.0865i·33-s + 0.344i·37-s − 0.153·39-s + 1.45·41-s − 0.284i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.332413709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.332413709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.364iT - 3T^{2} \) |
| 11 | \( 1 - 1.36T + 11T^{2} \) |
| 13 | \( 1 + 2.63iT - 13T^{2} \) |
| 17 | \( 1 + 2.23iT - 17T^{2} \) |
| 19 | \( 1 - 6.23T + 19T^{2} \) |
| 23 | \( 1 - 6.59iT - 23T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 - 4.50T + 31T^{2} \) |
| 37 | \( 1 - 2.09iT - 37T^{2} \) |
| 41 | \( 1 - 9.32T + 41T^{2} \) |
| 43 | \( 1 + 1.86iT - 43T^{2} \) |
| 47 | \( 1 - 6.86iT - 47T^{2} \) |
| 53 | \( 1 + 10.1iT - 53T^{2} \) |
| 59 | \( 1 - 1.63T + 59T^{2} \) |
| 61 | \( 1 - 0.0394T + 61T^{2} \) |
| 67 | \( 1 - 6.72iT - 67T^{2} \) |
| 71 | \( 1 + 6.27T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 5.32T + 79T^{2} \) |
| 83 | \( 1 + 14.7iT - 83T^{2} \) |
| 89 | \( 1 + 0.867T + 89T^{2} \) |
| 97 | \( 1 - 16.0iT - 97T^{2} \) |
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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
−7.961110135894256513115698731353, −7.48723539932694107368842994348, −6.96625111771330561175906049093, −5.96706815955617418156495751140, −5.35280329178595377497406234064, −4.50327173816246430862506249695, −3.63038317497340031331670348944, −2.88257816316135721973708036623, −1.67047870494008363905398141329, −0.839257652874225131349406403988,
0.952585756712267135441571777340, 1.90829768364687586449502377117, 2.97483603863308608325262032382, 4.03757705233398314129764897321, 4.39536991300093139827790212970, 5.38294370800787166816124174878, 6.19356072695149168790322282069, 6.94440953525055719110560565135, 7.49282471725118950373805469806, 8.321128527624665634334960361378
