| L(s) = 1 | + i·7-s − 3·11-s + i·13-s + 3i·17-s − 2·19-s − 6i·23-s − 9·29-s + 8·31-s − 10i·37-s − 2i·43-s + 3i·47-s − 49-s + 12·59-s + 8·61-s + 8i·67-s + ⋯ |
| L(s) = 1 | + 0.377i·7-s − 0.904·11-s + 0.277i·13-s + 0.727i·17-s − 0.458·19-s − 1.25i·23-s − 1.67·29-s + 1.43·31-s − 1.64i·37-s − 0.304i·43-s + 0.437i·47-s − 0.142·49-s + 1.56·59-s + 1.02·61-s + 0.977i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.472608557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.472608557\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 17iT - 97T^{2} \) |
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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
−7.995036569844709837871582790280, −7.36615797447158924154017608690, −6.49162882181503753863755006294, −5.88427825224716048352033220052, −5.16847347205826051430424425578, −4.35399722174269092243911263671, −3.61465765061589448717294852684, −2.52185719193018361747496175556, −1.98701765255630521234778684789, −0.50458200753257436089515812021,
0.75084371025582199638616592370, 1.94285307474004409515243035418, 2.87705427227391922406622209807, 3.61411530680295862166639123517, 4.54744527736157926293075129106, 5.24942638568418603780842995720, 5.86462622250061922150790372894, 6.83022625351355182490901155850, 7.36861687619956048761521613228, 8.097536460848612501766689778866
