| L(s) = 1 | − 2.75i·3-s + (2.03 + 0.928i)5-s − 4.29·7-s − 4.59·9-s + 3.89i·11-s − 1.27·13-s + (2.55 − 5.60i)15-s + 3.77i·17-s + (−4.27 + 0.851i)19-s + 11.8i·21-s + 1.36·23-s + (3.27 + 3.77i)25-s + 4.39i·27-s + 4.02i·29-s + 10.0·31-s + ⋯ |
| L(s) = 1 | − 1.59i·3-s + (0.909 + 0.415i)5-s − 1.62·7-s − 1.53·9-s + 1.17i·11-s − 0.353·13-s + (0.660 − 1.44i)15-s + 0.916i·17-s + (−0.980 + 0.195i)19-s + 2.58i·21-s + 0.284·23-s + (0.654 + 0.755i)25-s + 0.844i·27-s + 0.748i·29-s + 1.80·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9828922939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9828922939\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.03 - 0.928i)T \) |
| 19 | \( 1 + (4.27 - 0.851i)T \) |
| good | 3 | \( 1 + 2.75iT - 3T^{2} \) |
| 7 | \( 1 + 4.29T + 7T^{2} \) |
| 11 | \( 1 - 3.89iT - 11T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 - 3.77iT - 17T^{2} \) |
| 23 | \( 1 - 1.36T + 23T^{2} \) |
| 29 | \( 1 - 4.02iT - 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 8.39T + 37T^{2} \) |
| 41 | \( 1 - 6.20iT - 41T^{2} \) |
| 43 | \( 1 + 0.671T + 43T^{2} \) |
| 47 | \( 1 + 9.54T + 47T^{2} \) |
| 53 | \( 1 + 9.95T + 53T^{2} \) |
| 59 | \( 1 + 3.08T + 59T^{2} \) |
| 61 | \( 1 + 6.74T + 61T^{2} \) |
| 67 | \( 1 - 13.4iT - 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 1.28iT - 73T^{2} \) |
| 79 | \( 1 - 4.94T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 0.805iT - 89T^{2} \) |
| 97 | \( 1 - 1.15T + 97T^{2} \) |
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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
−9.773762100582010659221313853428, −8.713136780305143341576640289861, −7.75407504803150889931927626505, −6.93208021164123505925806109122, −6.33388103651054758047042533998, −6.11244155946010603882109537833, −4.62411609026573172742648827441, −3.09869105135382431991877030783, −2.38214213264734303706798544015, −1.39390047777222381829631046961,
0.37896076344505265105169810288, 2.67694021255423893126403990278, 3.25402403797746804449677082773, 4.37644248674752896448888462224, 5.09259569033397525476052132944, 6.15944115110234942350248511156, 6.42314872467357017718959612328, 8.056668311350557244529443426172, 9.119129484505133760961569910845, 9.325159131951662918621820971449
