L(s) = 1 | + (3.50 + 4.04i)2-s + (−4.18 − 2.69i)3-s + (−2.93 + 20.3i)4-s + (6.92 − 15.1i)5-s + (−3.78 − 26.3i)6-s + (7.38 + 2.16i)7-s + (−56.7 + 36.4i)8-s + (−0.923 − 2.02i)9-s + (85.5 − 25.1i)10-s + (−12.2 + 14.0i)11-s + (67.1 − 77.5i)12-s + (−28.3 + 8.33i)13-s + (17.0 + 37.4i)14-s + (−69.8 + 44.8i)15-s + (−187. − 55.1i)16-s + (−6.29 − 43.7i)17-s + ⋯ |
L(s) = 1 | + (1.23 + 1.42i)2-s + (−0.805 − 0.517i)3-s + (−0.366 + 2.54i)4-s + (0.619 − 1.35i)5-s + (−0.257 − 1.79i)6-s + (0.398 + 0.117i)7-s + (−2.50 + 1.61i)8-s + (−0.0342 − 0.0748i)9-s + (2.70 − 0.794i)10-s + (−0.334 + 0.386i)11-s + (1.61 − 1.86i)12-s + (−0.605 + 0.177i)13-s + (0.326 + 0.714i)14-s + (−1.20 + 0.772i)15-s + (−2.93 − 0.861i)16-s + (−0.0898 − 0.624i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.30572 + 0.957279i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.30572 + 0.957279i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 + (-104. - 34.4i)T \) |
good | 2 | \( 1 + (-3.50 - 4.04i)T + (-1.13 + 7.91i)T^{2} \) |
| 3 | \( 1 + (4.18 + 2.69i)T + (11.2 + 24.5i)T^{2} \) |
| 5 | \( 1 + (-6.92 + 15.1i)T + (-81.8 - 94.4i)T^{2} \) |
| 7 | \( 1 + (-7.38 - 2.16i)T + (288. + 185. i)T^{2} \) |
| 11 | \( 1 + (12.2 - 14.0i)T + (-189. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (28.3 - 8.33i)T + (1.84e3 - 1.18e3i)T^{2} \) |
| 17 | \( 1 + (6.29 + 43.7i)T + (-4.71e3 + 1.38e3i)T^{2} \) |
| 19 | \( 1 + (8.21 - 57.1i)T + (-6.58e3 - 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-18.6 - 129. i)T + (-2.34e4 + 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-128. + 82.3i)T + (1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 + (35.6 + 78.0i)T + (-3.31e4 + 3.82e4i)T^{2} \) |
| 41 | \( 1 + (-138. + 304. i)T + (-4.51e4 - 5.20e4i)T^{2} \) |
| 43 | \( 1 + (168. + 108. i)T + (3.30e4 + 7.23e4i)T^{2} \) |
| 47 | \( 1 - 126.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-728. - 213. i)T + (1.25e5 + 8.04e4i)T^{2} \) |
| 59 | \( 1 + (711. - 208. i)T + (1.72e5 - 1.11e5i)T^{2} \) |
| 61 | \( 1 + (221. - 142. i)T + (9.42e4 - 2.06e5i)T^{2} \) |
| 67 | \( 1 + (-66.9 - 77.2i)T + (-4.28e4 + 2.97e5i)T^{2} \) |
| 71 | \( 1 + (180. + 208. i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-86.8 + 604. i)T + (-3.73e5 - 1.09e5i)T^{2} \) |
| 79 | \( 1 + (954. - 280. i)T + (4.14e5 - 2.66e5i)T^{2} \) |
| 83 | \( 1 + (-154. - 337. i)T + (-3.74e5 + 4.32e5i)T^{2} \) |
| 89 | \( 1 + (-630. - 404. i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (153. - 337. i)T + (-5.97e5 - 6.89e5i)T^{2} \) |
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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
−17.13760284763005982427172328043, −16.52536951705759889638632310888, −15.12941844868796853326315484818, −13.76587191339510110715519441027, −12.68305953965038576644477047700, −12.00906610424892587299200246221, −8.891703527710865757662586729469, −7.25997703259715352777940620042, −5.72065386218414150525307691457, −4.84447338080152930440454015934,
2.72713363669187928373647225172, 4.82505251734873678412291096532, 6.20195830813757896180853743769, 10.07043371694425682489871986226, 10.77968735149823655078298974530, 11.56704313918216150222393406290, 13.23280933961364001035476405716, 14.31469679345750467250407657695, 15.28260594643182834865367188626, 17.40368678889550715077284197638