| L(s) = 1 | + i·2-s − 4-s + i·5-s + 7-s − i·8-s − 10-s − 3·11-s − 6i·13-s + i·14-s + 16-s + 3i·17-s + 6i·19-s − i·20-s − 3i·22-s − 6i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.447i·5-s + 0.377·7-s − 0.353i·8-s − 0.316·10-s − 0.904·11-s − 1.66i·13-s + 0.267i·14-s + 0.250·16-s + 0.727i·17-s + 1.37i·19-s − 0.223i·20-s − 0.639i·22-s − 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2778802160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2778802160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 37 | \( 1 + (-1 + 6i)T \) |
| good | 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 9iT - 29T^{2} \) |
| 31 | \( 1 - 3iT - 31T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 9iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 9iT - 61T^{2} \) |
| 67 | \( 1 + 14T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 - 3iT - 97T^{2} \) |
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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
−8.256045665020811152954229247989, −7.75686600491650076195366328942, −7.00725728676128280281215264287, −6.09904064115191180052452910688, −5.46753985966985822598172047265, −4.84044807002322263231879041209, −3.65593859238846918569404562188, −2.97150232130048190712512345387, −1.66695384825412519553343323703, −0.083806048222460538678434630380,
1.34102026886197044130013945807, 2.28135096276193073031715815484, 3.13573136308209404159706203882, 4.42962014701178618038349445039, 4.67826242140415900847189877807, 5.61892550125665444279650634245, 6.61851589649924298723094933432, 7.49885702635880212026379367073, 8.173057741731193378673411281362, 9.029169250814739430084650571497
