| L(s) = 1 | − 2i·3-s + 4i·7-s − 9-s − 4·11-s + 6i·17-s + 19-s + 8·21-s − 8i·23-s − 4i·27-s + 6·29-s − 8·31-s + 8i·33-s − 8i·37-s − 2·41-s + 12i·47-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + 1.51i·7-s − 0.333·9-s − 1.20·11-s + 1.45i·17-s + 0.229·19-s + 1.74·21-s − 1.66i·23-s − 0.769i·27-s + 1.11·29-s − 1.43·31-s + 1.39i·33-s − 1.31i·37-s − 0.312·41-s + 1.75i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.000460195487339548324177914263, −7.51071475565448399509113536047, −6.41514921022252884711175715544, −6.05096835650746317106816029269, −5.26867140498381639996719159591, −4.33871433569412497794628681530, −2.96404099878695769848514530656, −2.33780051628402365812606313704, −1.55884944833532194930608619431, 0,
1.38591285822656762466850296992, 2.96162197558000033173489606452, 3.52362534661250464524195810533, 4.45729880576578919058790671219, 4.97129473192671536790561627284, 5.65729213553092159166091263285, 7.04632849589007647778232078980, 7.32270154344778649384601505480, 8.126184110724334979399942197226
