| L(s) = 1 | − i·2-s + i·3-s − 4-s + (0.707 − 2.12i)5-s + 6-s + 2i·7-s + i·8-s − 9-s + (−2.12 − 0.707i)10-s + 4.24·11-s − i·12-s + 4.82i·13-s + 2·14-s + (2.12 + 0.707i)15-s + 16-s − 1.17i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.316 − 0.948i)5-s + 0.408·6-s + 0.755i·7-s + 0.353i·8-s − 0.333·9-s + (−0.670 − 0.223i)10-s + 1.27·11-s − 0.288i·12-s + 1.33i·13-s + 0.534·14-s + (0.547 + 0.182i)15-s + 0.250·16-s − 0.284i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.58258 - 0.256818i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58258 - 0.256818i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 + (-0.707 + 2.12i)T \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 - 4.82iT - 13T^{2} \) |
| 17 | \( 1 + 1.17iT - 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 3.41iT - 37T^{2} \) |
| 41 | \( 1 + 1.17T + 41T^{2} \) |
| 43 | \( 1 - 1.75iT - 43T^{2} \) |
| 47 | \( 1 + 4.82iT - 47T^{2} \) |
| 53 | \( 1 - 13.4iT - 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 + 3.41T + 61T^{2} \) |
| 67 | \( 1 - 0.585iT - 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 3.65iT - 73T^{2} \) |
| 79 | \( 1 + 7.65T + 79T^{2} \) |
| 83 | \( 1 + 1.41iT - 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−10.27693021387199110905159303060, −9.415932115066270440448047042429, −9.038493301593856911543272534412, −8.333299542889023975625281667380, −6.71683285618193204711957169424, −5.71823973354859464208238041088, −4.67086208995320879786151109404, −4.04468690662959987181201492212, −2.58579478179252669497453466095, −1.28810364690742197723188578930,
1.09905093290572192911390018785, 2.91712678565351579783689073692, 3.95168773788431369212375910229, 5.35001672789286272103546967679, 6.39984659361714939097998460042, 6.83621072564704550469791702038, 7.75245094452659092138341020588, 8.500884590332753668110169663426, 9.772163455233306341646731861350, 10.32087799143558851942097865662
