| L(s) = 1 | + (0.587 + 0.809i)2-s + (−1.31 − 0.428i)3-s + (−0.309 + 0.951i)4-s + (1.34 − 1.78i)5-s + (−0.428 − 1.31i)6-s − i·7-s + (−0.951 + 0.309i)8-s + (−0.872 − 0.633i)9-s + (2.23 + 0.0373i)10-s + (1.29 − 0.939i)11-s + (0.814 − 1.12i)12-s + (2.27 − 3.12i)13-s + (0.809 − 0.587i)14-s + (−2.53 + 1.77i)15-s + (−0.809 − 0.587i)16-s + (4.97 − 1.61i)17-s + ⋯ |
| L(s) = 1 | + (0.415 + 0.572i)2-s + (−0.761 − 0.247i)3-s + (−0.154 + 0.475i)4-s + (0.601 − 0.799i)5-s + (−0.174 − 0.538i)6-s − 0.377i·7-s + (−0.336 + 0.109i)8-s + (−0.290 − 0.211i)9-s + (0.707 + 0.0118i)10-s + (0.389 − 0.283i)11-s + (0.235 − 0.323i)12-s + (0.630 − 0.867i)13-s + (0.216 − 0.157i)14-s + (−0.655 + 0.459i)15-s + (−0.202 − 0.146i)16-s + (1.20 − 0.392i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30027 - 0.357089i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30027 - 0.357089i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.587 - 0.809i)T \) |
| 5 | \( 1 + (-1.34 + 1.78i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 + (1.31 + 0.428i)T + (2.42 + 1.76i)T^{2} \) |
| 11 | \( 1 + (-1.29 + 0.939i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.27 + 3.12i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.97 + 1.61i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.258 - 0.796i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.52 + 3.47i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.00 - 6.16i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.491 - 1.51i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.09 + 8.39i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.12 - 2.27i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 7.61iT - 43T^{2} \) |
| 47 | \( 1 + (-0.643 - 0.208i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (8.40 + 2.73i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (7.35 + 5.34i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (6.47 - 4.70i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (6.46 - 2.09i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (4.32 - 13.2i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.777 + 1.06i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (2.35 - 7.25i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.420 + 0.136i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-5.20 + 3.78i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-17.5 - 5.70i)T + (78.4 + 57.0i)T^{2} \) |
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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
−11.58757718118202409641404987919, −10.59764884306780972241388092651, −9.455735405757463663931690528904, −8.498987679623577566838762590421, −7.51699195133157463795172949089, −6.16148010800577379229493926535, −5.77306409172992139857369366185, −4.71613552218210819146157849682, −3.28869885394186121255267937053, −0.986517902176580942655392598398,
1.85233771816432312653582606450, 3.25937361401639959181444646479, 4.57084056944306929464097592883, 5.88760242090039279571366240495, 6.18526526183356112777026132204, 7.69883912246037150462904822774, 9.173568240505362886416250219266, 9.976758692933518791537830977435, 10.78550065031491501656460698682, 11.60520512300613374485768584707
