| L(s) = 1 | + 3-s + 3.46i·7-s + 9-s + 3.46i·11-s + 3.46i·13-s − 6·17-s + (−4 − 1.73i)19-s + 3.46i·21-s − 5·25-s + 27-s − 6.92i·29-s + 10·31-s + 3.46i·33-s + 3.46i·37-s + 3.46i·39-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.30i·7-s + 0.333·9-s + 1.04i·11-s + 0.960i·13-s − 1.45·17-s + (−0.917 − 0.397i)19-s + 0.755i·21-s − 25-s + 0.192·27-s − 1.28i·29-s + 1.79·31-s + 0.603i·33-s + 0.569i·37-s + 0.554i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04315 + 1.17053i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04315 + 1.17053i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + (4 + 1.73i)T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 - 3.46iT - 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 - 6.92iT - 47T^{2} \) |
| 53 | \( 1 + 13.8iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−9.928893733422083180700432704864, −9.533102319122955538459747419432, −8.599715891584782800875989555332, −8.068998870698734215836816043440, −6.74179197046304684145964181699, −6.25390693417495342199817470878, −4.79120571923616587209650478490, −4.20194994975886507588875225753, −2.54561658148014300309382015204, −2.04358279692190287755073988092,
0.66959445992424545105962945263, 2.30900102770700837006684079715, 3.58786659845688741694869860454, 4.21364101527233722140472561117, 5.50037755594958602220893982706, 6.61005108446393598455291668793, 7.33865216613293070777849546123, 8.356909299591533214093019428851, 8.763484474856935053271925669620, 10.08334375561979251765025757226
