| L(s) = 1 | + (14.0 + 11.7i)3-s + (57.8 − 21.0i)5-s + (68.1 + 117. i)7-s + (15.9 + 90.2i)9-s + (211. − 366. i)11-s + (1.68 − 1.41i)13-s + (1.05e3 + 385. i)15-s + (−204. + 1.16e3i)17-s + (−1.57e3 + 36.6i)19-s + (−432. + 2.45e3i)21-s + (4.19e3 + 1.52e3i)23-s + (512. − 430. i)25-s + (1.38e3 − 2.39e3i)27-s + (153. + 868. i)29-s + (1.29e3 + 2.23e3i)31-s + ⋯ |
| L(s) = 1 | + (0.898 + 0.754i)3-s + (1.03 − 0.376i)5-s + (0.525 + 0.909i)7-s + (0.0655 + 0.371i)9-s + (0.527 − 0.913i)11-s + (0.00276 − 0.00231i)13-s + (1.21 + 0.442i)15-s + (−0.171 + 0.975i)17-s + (−0.999 + 0.0233i)19-s + (−0.214 + 1.21i)21-s + (1.65 + 0.601i)23-s + (0.164 − 0.137i)25-s + (0.365 − 0.632i)27-s + (0.0338 + 0.191i)29-s + (0.241 + 0.417i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.74394 + 1.01039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.74394 + 1.01039i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.57e3 - 36.6i)T \) |
| good | 3 | \( 1 + (-14.0 - 11.7i)T + (42.1 + 239. i)T^{2} \) |
| 5 | \( 1 + (-57.8 + 21.0i)T + (2.39e3 - 2.00e3i)T^{2} \) |
| 7 | \( 1 + (-68.1 - 117. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-211. + 366. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-1.68 + 1.41i)T + (6.44e4 - 3.65e5i)T^{2} \) |
| 17 | \( 1 + (204. - 1.16e3i)T + (-1.33e6 - 4.85e5i)T^{2} \) |
| 23 | \( 1 + (-4.19e3 - 1.52e3i)T + (4.93e6 + 4.13e6i)T^{2} \) |
| 29 | \( 1 + (-153. - 868. i)T + (-1.92e7 + 7.01e6i)T^{2} \) |
| 31 | \( 1 + (-1.29e3 - 2.23e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + 3.29e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (4.05e3 + 3.40e3i)T + (2.01e7 + 1.14e8i)T^{2} \) |
| 43 | \( 1 + (1.27e4 - 4.63e3i)T + (1.12e8 - 9.44e7i)T^{2} \) |
| 47 | \( 1 + (3.52e3 + 1.99e4i)T + (-2.15e8 + 7.84e7i)T^{2} \) |
| 53 | \( 1 + (2.41e4 + 8.78e3i)T + (3.20e8 + 2.68e8i)T^{2} \) |
| 59 | \( 1 + (-3.32e3 + 1.88e4i)T + (-6.71e8 - 2.44e8i)T^{2} \) |
| 61 | \( 1 + (4.67e3 + 1.70e3i)T + (6.46e8 + 5.42e8i)T^{2} \) |
| 67 | \( 1 + (8.52e3 + 4.83e4i)T + (-1.26e9 + 4.61e8i)T^{2} \) |
| 71 | \( 1 + (2.45e4 - 8.92e3i)T + (1.38e9 - 1.15e9i)T^{2} \) |
| 73 | \( 1 + (-4.99e4 - 4.19e4i)T + (3.59e8 + 2.04e9i)T^{2} \) |
| 79 | \( 1 + (6.87e4 + 5.76e4i)T + (5.34e8 + 3.03e9i)T^{2} \) |
| 83 | \( 1 + (2.86e4 + 4.97e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (2.87e4 - 2.40e4i)T + (9.69e8 - 5.49e9i)T^{2} \) |
| 97 | \( 1 + (-2.60e3 + 1.47e4i)T + (-8.06e9 - 2.93e9i)T^{2} \) |
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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
−13.81891707775222115697758909050, −12.77952363852739580550131926623, −11.32152026922898637708487146370, −10.04332508151663062241045238951, −8.868190627102781523627980659575, −8.577727483686391620049392139604, −6.28980626237790246892938520247, −5.04175747815629128487185165777, −3.36325778427458976711500417757, −1.79925885489647097054106251128,
1.45238718647505571327963246469, 2.64518051149550795613001044842, 4.65904375655854068052767968185, 6.62705165164124450385893583394, 7.43602882058774646116725166118, 8.789049791256704673700178121918, 9.946582817398019020032038815673, 11.08915779004003089581334650439, 12.70086641065420474058208932908, 13.59113036604234437288241211799
