| L(s) = 1 | + 2i·5-s + (2.23 + 1.41i)7-s − 1.41·11-s + 4.47·17-s + 6.32·19-s + 3.16i·23-s + 25-s − 3.16·29-s − 5.65i·31-s + (−2.82 + 4.47i)35-s + 4.47i·37-s + 4.47·41-s − 4i·43-s + 8·47-s + (3.00 + 6.32i)49-s + ⋯ |
| L(s) = 1 | + 0.894i·5-s + (0.845 + 0.534i)7-s − 0.426·11-s + 1.08·17-s + 1.45·19-s + 0.659i·23-s + 0.200·25-s − 0.587·29-s − 1.01i·31-s + (−0.478 + 0.755i)35-s + 0.735i·37-s + 0.698·41-s − 0.609i·43-s + 1.16·47-s + (0.428 + 0.903i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.259505124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.259505124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.23 - 1.41i)T \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 23 | \( 1 - 3.16iT - 23T^{2} \) |
| 29 | \( 1 + 3.16T + 29T^{2} \) |
| 31 | \( 1 + 5.65iT - 31T^{2} \) |
| 37 | \( 1 - 4.47iT - 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 9.48T + 53T^{2} \) |
| 59 | \( 1 + 8.94iT - 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 9.48iT - 71T^{2} \) |
| 73 | \( 1 + 6.32iT - 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 6.32iT - 97T^{2} \) |
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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
−8.447414206274547472498593800019, −7.63663119046530136787542776158, −7.41858596385834477728522048105, −6.32124387928290515137987325875, −5.50358649256593844225482155777, −5.07905074334282927462465850701, −3.85448320516047175546696792187, −3.06416911732364035769728536637, −2.27695709997569167446503436784, −1.12767580719912969012009912035,
0.78535435817670845648564722556, 1.50599181665335712718699869826, 2.79767189411040286462902307975, 3.77526365379041677770651815138, 4.67580478282357054250349901537, 5.22923984808884574072963746116, 5.84950751936036500188785614393, 7.10289530367415521714471202957, 7.62527755408726812260824318547, 8.245892476968189091518968699330
