| L(s) = 1 | − 2.28i·3-s − 2.23·5-s − 2.82i·7-s − 2.23·9-s + 5.23·11-s − 4.03i·13-s + 5.11i·15-s − 1.08i·17-s − 19-s − 6.47·21-s − 7.40i·23-s + 5.00·25-s − 1.74i·27-s − 4.47·29-s + 4·31-s + ⋯ |
| L(s) = 1 | − 1.32i·3-s − 0.999·5-s − 1.06i·7-s − 0.745·9-s + 1.57·11-s − 1.11i·13-s + 1.32i·15-s − 0.262i·17-s − 0.229·19-s − 1.41·21-s − 1.54i·23-s + 1.00·25-s − 0.336i·27-s − 0.830·29-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.244546928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.244546928\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.28iT - 3T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 + 4.03iT - 13T^{2} \) |
| 17 | \( 1 + 1.08iT - 17T^{2} \) |
| 23 | \( 1 + 7.40iT - 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 6.86iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 8.48iT - 47T^{2} \) |
| 53 | \( 1 - 6.86iT - 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 1.70T + 61T^{2} \) |
| 67 | \( 1 + 1.62iT - 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 - 13.7iT - 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 5.24iT - 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 - 4.44iT - 97T^{2} \) |
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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
−8.713406643219459394982778866512, −8.112101416928282065364709412733, −7.39393990544659752977321113716, −6.78600600103757604026136685597, −6.21733726424002316977766055676, −4.69478788221482684592070565224, −3.92299089664035399741985989539, −2.91104047446450596028450771723, −1.35922589834201333082967658028, −0.54542025728583361434968334480,
1.77219049009842438974043788465, 3.37792996068683155764169322055, 3.93149059235761268457382891037, 4.64367850574787660840022587846, 5.62069275659300010840208802164, 6.58398322769346663071247319122, 7.49506399577974839075843619224, 8.664152288645438628781776669386, 9.142311926297458592004019841102, 9.569351058530270919248124202637
