| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s + 3i·7-s − i·8-s + 2·9-s + 2·11-s − i·12-s + i·13-s − 3·14-s + 16-s + 3i·17-s + 2i·18-s + 19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.13i·7-s − 0.353i·8-s + 0.666·9-s + 0.603·11-s − 0.288i·12-s + 0.277i·13-s − 0.801·14-s + 0.250·16-s + 0.727i·17-s + 0.471i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.342341 + 1.45018i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.342341 + 1.45018i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 + 15T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 9iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−10.20686177070777993702787758356, −9.290777812594767207445269516254, −8.921527482239301520337262919554, −7.908104386178803227334811603934, −6.92435317740451826562152346022, −6.08546600481128978129866741152, −5.22989644326966415173770998739, −4.33044112260669478401994054540, −3.35882232367253671756552109964, −1.73532994630650044859878851120,
0.75062600675756926202476192265, 1.79230948075124349973494254521, 3.24569016120247373212380306101, 4.16604537503819153879657329221, 5.06327090117280067815267076119, 6.47540811791126218630180644646, 7.18071432601313327592947201680, 7.919176603635043067459351098544, 8.993274606774424378511424752687, 9.896540214040990582267120060212
