| L(s) = 1 | + (3.67 + 2.67i)2-s + (−0.927 − 2.85i)3-s + (3.91 + 12.0i)4-s + (6.73 + 8.92i)5-s + (4.21 − 12.9i)6-s + 9.62·7-s + (−6.54 + 20.1i)8-s + (−7.28 + 5.29i)9-s + (0.908 + 50.8i)10-s + (−4.49 − 3.26i)11-s + (30.7 − 22.3i)12-s + (−39.0 + 28.3i)13-s + (35.4 + 25.7i)14-s + (19.2 − 27.4i)15-s + (4.03 − 2.92i)16-s + (29.0 − 89.3i)17-s + ⋯ |
| L(s) = 1 | + (1.30 + 0.944i)2-s + (−0.178 − 0.549i)3-s + (0.489 + 1.50i)4-s + (0.602 + 0.798i)5-s + (0.286 − 0.882i)6-s + 0.519·7-s + (−0.289 + 0.890i)8-s + (−0.269 + 0.195i)9-s + (0.0287 + 1.60i)10-s + (−0.123 − 0.0895i)11-s + (0.739 − 0.537i)12-s + (−0.832 + 0.605i)13-s + (0.676 + 0.491i)14-s + (0.330 − 0.473i)15-s + (0.0629 − 0.0457i)16-s + (0.414 − 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.40695 + 1.58858i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.40695 + 1.58858i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.927 + 2.85i)T \) |
| 5 | \( 1 + (-6.73 - 8.92i)T \) |
| good | 2 | \( 1 + (-3.67 - 2.67i)T + (2.47 + 7.60i)T^{2} \) |
| 7 | \( 1 - 9.62T + 343T^{2} \) |
| 11 | \( 1 + (4.49 + 3.26i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (39.0 - 28.3i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-29.0 + 89.3i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-14.5 + 44.6i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (143. + 104. i)T + (3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (20.1 + 61.8i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-37.2 + 114. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (6.32 - 4.59i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (390. - 283. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 421.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-134. - 414. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-4.44 - 13.6i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (430. - 312. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-380. - 276. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-81.8 + 251. i)T + (-2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-60.2 - 185. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (66.1 + 48.0i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-379. - 1.16e3i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-329. + 1.01e3i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (941. + 683. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-351. - 1.08e3i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−14.13266348467045208221513595518, −13.63277787783074056644292262926, −12.33995047203763639836070763302, −11.39174260604409999850223554343, −9.772450562993552644820196370387, −7.77971418113166063670340978080, −6.89419792766851483960188217057, −5.86893901135965414084830727167, −4.63983527616140013043557878705, −2.63512541549245277615230536164,
1.83596395778117514518332950799, 3.74251143498605475735763857487, 5.03956444631272057157292637860, 5.77305852221869557085466064338, 8.200870016597521017331751293816, 9.866653134807516236052665495666, 10.61845861941733962320167488209, 12.01683828506411270625412159373, 12.56354148720396175425178798599, 13.75616904141857597395815445295
