| L(s) = 1 | + 2i·2-s − 2·4-s − 3i·7-s + 11-s − i·13-s + 6·14-s − 4·16-s − i·17-s + 2·19-s + 2i·22-s + 3i·23-s + 2·26-s + 6i·28-s − 2·29-s − 6·31-s − 8i·32-s + ⋯ |
| L(s) = 1 | + 1.41i·2-s − 4-s − 1.13i·7-s + 0.301·11-s − 0.277i·13-s + 1.60·14-s − 16-s − 0.242i·17-s + 0.458·19-s + 0.426i·22-s + 0.625i·23-s + 0.392·26-s + 1.13i·28-s − 0.371·29-s − 1.07·31-s − 1.41i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.549107845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.549107845\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 - 2iT - 2T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 11iT - 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 10iT - 47T^{2} \) |
| 53 | \( 1 + 11iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 5T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 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
−8.538322671737764831684737668083, −7.84435811853760561171427918230, −7.07978112278912470948974670538, −6.92908566360159277454193116003, −5.70001926955390453610685032465, −5.31213003662936222885185532060, −4.20855102704911379404082232144, −3.55103952631971567866707456847, −2.06356156990502496368370980074, −0.54684343669856144119168045931,
1.11616088887264556073464043862, 2.13667853204613269870672906780, 2.81377589737374436834965191329, 3.71648278380554459268328947114, 4.55577859397052027099335077742, 5.50068394414761151734445971764, 6.32810379195930940007664517600, 7.19744131751515319901032682505, 8.269985654050410196391348434937, 9.037008400611670573419571767770
