| L(s) = 1 | + (−0.998 + 0.0475i)2-s + (1.35 − 1.07i)3-s + (0.995 − 0.0950i)4-s + (1.12 + 1.06i)5-s + (−1.30 + 1.13i)6-s + (−0.176 − 0.914i)7-s + (−0.989 + 0.142i)8-s + (0.688 − 2.91i)9-s + (−1.17 − 1.01i)10-s + (4.91 − 2.53i)11-s + (1.24 − 1.19i)12-s + (−6.08 − 1.17i)13-s + (0.219 + 0.905i)14-s + (2.67 + 0.246i)15-s + (0.981 − 0.189i)16-s + (1.67 + 3.67i)17-s + ⋯ |
| L(s) = 1 | + (−0.706 + 0.0336i)2-s + (0.784 − 0.620i)3-s + (0.497 − 0.0475i)4-s + (0.501 + 0.478i)5-s + (−0.532 + 0.464i)6-s + (−0.0666 − 0.345i)7-s + (−0.349 + 0.0503i)8-s + (0.229 − 0.973i)9-s + (−0.370 − 0.321i)10-s + (1.48 − 0.763i)11-s + (0.360 − 0.346i)12-s + (−1.68 − 0.325i)13-s + (0.0586 + 0.241i)14-s + (0.690 + 0.0636i)15-s + (0.245 − 0.0473i)16-s + (0.407 + 0.891i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33351 - 0.528547i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33351 - 0.528547i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.998 - 0.0475i)T \) |
| 3 | \( 1 + (-1.35 + 1.07i)T \) |
| 23 | \( 1 + (-3.86 + 2.83i)T \) |
| good | 5 | \( 1 + (-1.12 - 1.06i)T + (0.237 + 4.99i)T^{2} \) |
| 7 | \( 1 + (0.176 + 0.914i)T + (-6.49 + 2.60i)T^{2} \) |
| 11 | \( 1 + (-4.91 + 2.53i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (6.08 + 1.17i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (-1.67 - 3.67i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-3.84 - 1.75i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.430 - 4.50i)T + (-28.4 - 5.48i)T^{2} \) |
| 31 | \( 1 + (6.48 - 5.10i)T + (7.30 - 30.1i)T^{2} \) |
| 37 | \( 1 + (-1.24 + 4.25i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-4.63 + 4.86i)T + (-1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (1.83 - 2.33i)T + (-10.1 - 41.7i)T^{2} \) |
| 47 | \( 1 + (1.91 + 1.10i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.08 + 1.24i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (1.84 - 9.57i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (1.04 - 2.60i)T + (-44.1 - 42.0i)T^{2} \) |
| 67 | \( 1 + (-3.62 + 7.03i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (-1.75 - 2.72i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (5.81 - 12.7i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (3.68 - 1.27i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (6.53 - 6.22i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (1.97 - 13.7i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (13.3 + 3.22i)T + (86.2 + 44.4i)T^{2} \) |
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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
−10.91454766038731796137916000867, −9.988050943550519302877913083057, −9.255810386195974966491323362537, −8.448613384506703397301229699768, −7.30831154222670765884489998332, −6.82325867077141957731047478054, −5.69112359411797870036611321012, −3.74315483363315806564278990108, −2.64189759864669302338480820239, −1.25668991277301960989846555699,
1.74114544917768514327491998789, 2.95214786209499255779085458369, 4.46263665969946064620402986351, 5.43092061115122657974910783706, 7.04709215330575889207520338972, 7.66005983806055583703085632660, 9.088174220658776531486432569326, 9.561958130702373474980095566715, 9.713525362596242398688224005818, 11.31968557987534009043834478497
