| L(s) = 1 | + (−1.79 + 1.15i)2-s + (2.21 − 15.4i)3-s + (−24.7 + 54.0i)4-s + (56.4 + 192. i)5-s + (13.7 + 30.1i)6-s + (−291. + 252. i)7-s + (−37.4 − 260. i)8-s + (−233. − 68.4i)9-s + (−322. − 279. i)10-s + (1.29e3 − 2.00e3i)11-s + (779. + 501. i)12-s + (−492. + 568. i)13-s + (231. − 788. i)14-s + (3.09e3 − 444. i)15-s + (−2.12e3 − 2.45e3i)16-s + (−7.99e3 + 3.65e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.223 + 0.143i)2-s + (0.0821 − 0.571i)3-s + (−0.385 + 0.845i)4-s + (0.451 + 1.53i)5-s + (0.0638 + 0.139i)6-s + (−0.850 + 0.736i)7-s + (−0.0730 − 0.508i)8-s + (−0.319 − 0.0939i)9-s + (−0.322 − 0.279i)10-s + (0.969 − 1.50i)11-s + (0.451 + 0.290i)12-s + (−0.224 + 0.258i)13-s + (0.0843 − 0.287i)14-s + (0.915 − 0.131i)15-s + (−0.518 − 0.598i)16-s + (−1.62 + 0.743i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.341i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0759008 - 0.431196i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0759008 - 0.431196i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.21 + 15.4i)T \) |
| 23 | \( 1 + (1.69e3 + 1.20e4i)T \) |
| good | 2 | \( 1 + (1.79 - 1.15i)T + (26.5 - 58.2i)T^{2} \) |
| 5 | \( 1 + (-56.4 - 192. i)T + (-1.31e4 + 8.44e3i)T^{2} \) |
| 7 | \( 1 + (291. - 252. i)T + (1.67e4 - 1.16e5i)T^{2} \) |
| 11 | \( 1 + (-1.29e3 + 2.00e3i)T + (-7.35e5 - 1.61e6i)T^{2} \) |
| 13 | \( 1 + (492. - 568. i)T + (-6.86e5 - 4.77e6i)T^{2} \) |
| 17 | \( 1 + (7.99e3 - 3.65e3i)T + (1.58e7 - 1.82e7i)T^{2} \) |
| 19 | \( 1 + (-705. - 322. i)T + (3.08e7 + 3.55e7i)T^{2} \) |
| 29 | \( 1 + (-1.16e4 - 2.55e4i)T + (-3.89e8 + 4.49e8i)T^{2} \) |
| 31 | \( 1 + (5.30e3 + 3.68e4i)T + (-8.51e8 + 2.50e8i)T^{2} \) |
| 37 | \( 1 + (-1.66e3 + 5.68e3i)T + (-2.15e9 - 1.38e9i)T^{2} \) |
| 41 | \( 1 + (8.36e4 - 2.45e4i)T + (3.99e9 - 2.56e9i)T^{2} \) |
| 43 | \( 1 + (9.43e4 + 1.35e4i)T + (6.06e9 + 1.78e9i)T^{2} \) |
| 47 | \( 1 - 4.19e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (7.99e4 - 6.92e4i)T + (3.15e9 - 2.19e10i)T^{2} \) |
| 59 | \( 1 + (3.37e4 - 3.88e4i)T + (-6.00e9 - 4.17e10i)T^{2} \) |
| 61 | \( 1 + (-1.98e5 + 2.85e4i)T + (4.94e10 - 1.45e10i)T^{2} \) |
| 67 | \( 1 + (1.50e5 + 2.34e5i)T + (-3.75e10 + 8.22e10i)T^{2} \) |
| 71 | \( 1 + (2.21e5 - 1.42e5i)T + (5.32e10 - 1.16e11i)T^{2} \) |
| 73 | \( 1 + (-1.61e5 + 3.54e5i)T + (-9.91e10 - 1.14e11i)T^{2} \) |
| 79 | \( 1 + (-5.20e5 - 4.51e5i)T + (3.45e10 + 2.40e11i)T^{2} \) |
| 83 | \( 1 + (1.87e5 - 6.39e5i)T + (-2.75e11 - 1.76e11i)T^{2} \) |
| 89 | \( 1 + (1.57e5 + 2.27e4i)T + (4.76e11 + 1.40e11i)T^{2} \) |
| 97 | \( 1 + (-3.27e5 - 1.11e6i)T + (-7.00e11 + 4.50e11i)T^{2} \) |
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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
−13.95717545325582238673400525392, −13.18011485082370094892652650138, −11.94586353489075618064766303999, −10.86260802535202699509254620751, −9.306824222857360650382745941816, −8.411663800423461022058062892757, −6.70753665019969655827865965884, −6.33355793693592034535795493479, −3.56886671865131292123585712837, −2.52882252301841240216989784548,
0.17944868407949747245180996617, 1.65702328864693879820059974187, 4.29741470455643061691386329087, 5.09227102264004242281710518589, 6.72757946561867556730832661287, 8.789415433094614922074857597067, 9.564375561502310879408576616773, 10.10064652772240045204090615928, 11.78271400270397070020203965032, 13.10577963061174261346961885863
