| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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
−8.394470278633472419530094723645, −7.76889568611270854084961698475, −6.92132804975271915966961599000, −6.50610868637636646917078830835, −5.52758449138390867802516203492, −4.62794525228551877864348259884, −3.92575506650289542527979777879, −3.00933100975332601936715494727, −2.06657461874434545597828514865, −0.938751701066292678368803405720,
0.866072096041394515205549784064, 1.79534181303822149971383920451, 2.77068895540799566649481538220, 4.08541704598248660417838200688, 4.51758712360080033711656965694, 5.55302148365862356775414332941, 5.88354427190383825817160404435, 7.02710717726986848849146186746, 7.84800698207884968647882052628, 8.419307555099325232511040990637
