| L(s) = 1 | + (−0.926 − 0.376i)2-s + (−1.92 − 0.430i)3-s + (0.716 + 0.697i)4-s + (0.0390 + 1.41i)5-s + (1.61 + 1.12i)6-s + (−0.323 − 1.27i)7-s + (−0.401 − 0.915i)8-s + (0.796 + 0.375i)9-s + (0.496 − 1.32i)10-s + (0.0787 − 0.950i)11-s + (−1.07 − 1.64i)12-s + (1.17 + 3.76i)13-s + (−0.180 + 1.30i)14-s + (0.534 − 2.73i)15-s + (0.0275 + 0.999i)16-s + (−3.40 + 3.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.655 − 0.266i)2-s + (−1.10 − 0.248i)3-s + (0.358 + 0.348i)4-s + (0.0174 + 0.632i)5-s + (0.660 + 0.458i)6-s + (−0.122 − 0.482i)7-s + (−0.142 − 0.323i)8-s + (0.265 + 0.125i)9-s + (0.156 − 0.419i)10-s + (0.0237 − 0.286i)11-s + (−0.311 − 0.476i)12-s + (0.326 + 1.04i)13-s + (−0.0483 + 0.348i)14-s + (0.138 − 0.706i)15-s + (0.00688 + 0.249i)16-s + (−0.826 + 0.803i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.330420 - 0.367416i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.330420 - 0.367416i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.926 + 0.376i)T \) |
| 19 | \( 1 + (2.17 + 3.77i)T \) |
| good | 3 | \( 1 + (1.92 + 0.430i)T + (2.71 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.0390 - 1.41i)T + (-4.99 + 0.275i)T^{2} \) |
| 7 | \( 1 + (0.323 + 1.27i)T + (-6.15 + 3.33i)T^{2} \) |
| 11 | \( 1 + (-0.0787 + 0.950i)T + (-10.8 - 1.81i)T^{2} \) |
| 13 | \( 1 + (-1.17 - 3.76i)T + (-10.6 + 7.40i)T^{2} \) |
| 17 | \( 1 + (3.40 - 3.31i)T + (0.468 - 16.9i)T^{2} \) |
| 23 | \( 1 + (-6.07 + 1.36i)T + (20.8 - 9.81i)T^{2} \) |
| 29 | \( 1 + (0.812 + 0.229i)T + (24.7 + 15.1i)T^{2} \) |
| 31 | \( 1 + (4.41 + 3.43i)T + (7.61 + 30.0i)T^{2} \) |
| 37 | \( 1 + (-0.770 + 9.30i)T + (-36.4 - 6.08i)T^{2} \) |
| 41 | \( 1 + (0.437 + 0.381i)T + (5.63 + 40.6i)T^{2} \) |
| 43 | \( 1 + (3.09 + 8.27i)T + (-32.4 + 28.2i)T^{2} \) |
| 47 | \( 1 + (-5.69 - 3.94i)T + (16.4 + 44.0i)T^{2} \) |
| 53 | \( 1 + (9.23 + 6.40i)T + (18.5 + 49.6i)T^{2} \) |
| 59 | \( 1 + (6.15 + 5.36i)T + (8.10 + 58.4i)T^{2} \) |
| 61 | \( 1 + (4.41 + 6.01i)T + (-18.2 + 58.2i)T^{2} \) |
| 67 | \( 1 + (-6.23 - 0.689i)T + (65.3 + 14.6i)T^{2} \) |
| 71 | \( 1 + (-1.36 + 1.86i)T + (-21.1 - 67.7i)T^{2} \) |
| 73 | \( 1 + (-3.21 + 3.12i)T + (2.01 - 72.9i)T^{2} \) |
| 79 | \( 1 + (2.99 + 7.99i)T + (-59.5 + 51.8i)T^{2} \) |
| 83 | \( 1 + (-11.8 + 6.39i)T + (45.3 - 69.4i)T^{2} \) |
| 89 | \( 1 + (9.85 + 9.58i)T + (2.45 + 88.9i)T^{2} \) |
| 97 | \( 1 + (0.370 - 0.0410i)T + (94.6 - 21.2i)T^{2} \) |
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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
−10.67782992771497735317968325707, −9.256824799604896997091017726273, −8.697848423959917658517070386637, −7.26941453239384370957132963721, −6.72741067999316993416915280320, −6.05991268097138495357306546934, −4.71032699059950561710833273837, −3.51002277961995348916074955286, −2.05390433045303145452279354468, −0.43075420475728693791976527326,
1.12069167832730793286813989477, 2.88998936920590687716126596574, 4.67164795806374632358043077007, 5.34477803678258056868417636565, 6.13888594954926014492622977359, 7.05784909789482468020900863875, 8.218073606770857463449179988524, 8.918720910975870321954807348917, 9.768809990608117566349227668366, 10.74510245704767728037778570590
