| L(s) = 1 | + (−1.67 + 2.90i)2-s + (0.949 − 5.10i)3-s + (−1.62 − 2.80i)4-s − 8.70·5-s + (13.2 + 11.3i)6-s + (−14.4 − 11.6i)7-s − 15.9·8-s + (−25.1 − 9.70i)9-s + (14.5 − 25.2i)10-s − 11.7·11-s + (−15.8 + 5.61i)12-s + (26.5 − 46.0i)13-s + (57.9 − 22.3i)14-s + (−8.27 + 44.4i)15-s + (39.7 − 68.7i)16-s + (−22.5 + 39.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.592 + 1.02i)2-s + (0.182 − 0.983i)3-s + (−0.202 − 0.351i)4-s − 0.778·5-s + (0.901 + 0.770i)6-s + (−0.777 − 0.628i)7-s − 0.704·8-s + (−0.933 − 0.359i)9-s + (0.461 − 0.799i)10-s − 0.323·11-s + (−0.382 + 0.135i)12-s + (0.567 − 0.982i)13-s + (1.10 − 0.425i)14-s + (−0.142 + 0.765i)15-s + (0.620 − 1.07i)16-s + (−0.321 + 0.557i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.916i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.401 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.177543 - 0.271570i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.177543 - 0.271570i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.949 + 5.10i)T \) |
| 7 | \( 1 + (14.4 + 11.6i)T \) |
| good | 2 | \( 1 + (1.67 - 2.90i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 + 8.70T + 125T^{2} \) |
| 11 | \( 1 + 11.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-26.5 + 46.0i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (22.5 - 39.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (13.0 + 22.6i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + 58.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-36.7 - 63.5i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-157. - 272. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (189. + 327. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-181. + 313. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-58.7 - 101. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (114. - 197. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-307. + 532. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (288. + 499. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (223. - 386. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (148. + 257. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 866.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-283. + 490. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-183. + 317. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (510. + 884. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (247. + 427. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (76.3 + 132. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.24939522267438888690639617297, −13.00119205928674347644897047273, −12.13877449356103965538445185378, −10.59306948014863102339991990205, −8.839339795109992506741173007783, −7.920348024872233299853363760730, −7.06973593667816438209088069072, −5.99001314805621212658695888707, −3.33043782246892052000213610280, −0.25005024498551108835973439443,
2.69016040498687953432634143912, 4.08437899984512743927346482824, 6.09474537760091967786717060620, 8.316228627261782676538190108471, 9.333814674594410852181809156653, 10.15617115348282631954040595666, 11.42650606836207809336958094747, 11.96121699836210650167183333515, 13.61779876818172998059298989451, 15.21043336813923711472420029325
