| L(s) = 1 | + (3.73 + 2.39i)2-s + (−0.655 − 4.55i)3-s + (4.86 + 10.6i)4-s + (−9.99 − 2.93i)5-s + (8.49 − 18.5i)6-s + (−17.7 + 20.4i)7-s + (−2.34 + 16.2i)8-s + (5.55 − 1.62i)9-s + (−30.2 − 34.9i)10-s + (43.3 − 27.8i)11-s + (45.3 − 29.1i)12-s + (4.59 + 5.30i)13-s + (−115. + 33.9i)14-s + (−6.82 + 47.4i)15-s + (13.5 − 15.5i)16-s + (−44.2 + 96.9i)17-s + ⋯ |
| L(s) = 1 | + (1.32 + 0.848i)2-s + (−0.126 − 0.877i)3-s + (0.607 + 1.33i)4-s + (−0.893 − 0.262i)5-s + (0.577 − 1.26i)6-s + (−0.958 + 1.10i)7-s + (−0.103 + 0.719i)8-s + (0.205 − 0.0603i)9-s + (−0.957 − 1.10i)10-s + (1.18 − 0.764i)11-s + (1.09 − 0.701i)12-s + (0.0980 + 0.113i)13-s + (−2.20 + 0.647i)14-s + (−0.117 + 0.817i)15-s + (0.211 − 0.243i)16-s + (−0.631 + 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.69772 + 0.415920i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69772 + 0.415920i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-60.7 - 92.0i)T \) |
| good | 2 | \( 1 + (-3.73 - 2.39i)T + (3.32 + 7.27i)T^{2} \) |
| 3 | \( 1 + (0.655 + 4.55i)T + (-25.9 + 7.60i)T^{2} \) |
| 5 | \( 1 + (9.99 + 2.93i)T + (105. + 67.5i)T^{2} \) |
| 7 | \( 1 + (17.7 - 20.4i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (-43.3 + 27.8i)T + (552. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (-4.59 - 5.30i)T + (-312. + 2.17e3i)T^{2} \) |
| 17 | \( 1 + (44.2 - 96.9i)T + (-3.21e3 - 3.71e3i)T^{2} \) |
| 19 | \( 1 + (31.4 + 68.9i)T + (-4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (51.2 - 112. i)T + (-1.59e4 - 1.84e4i)T^{2} \) |
| 31 | \( 1 + (-7.80 + 54.2i)T + (-2.85e4 - 8.39e3i)T^{2} \) |
| 37 | \( 1 + (74.5 - 21.8i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (274. + 80.5i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (24.4 + 170. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 - 450.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (58.7 - 67.7i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-380. - 438. i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (-15.4 + 107. i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (43.6 + 28.0i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (368. + 236. i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-317. - 695. i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (372. + 429. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-363. + 106. i)T + (4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (25.8 + 179. i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 + (554. + 162. i)T + (7.67e5 + 4.93e5i)T^{2} \) |
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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
−17.07215320699627039981465819940, −15.76679563635218903650709344221, −15.13202290841916182138524646050, −13.47942795311239160177465574252, −12.59622402517979706152999434722, −11.76845817536015725817637668320, −8.764334586319264439514296577475, −6.97070124282006701310850193518, −6.00187086280397503243606870586, −3.82168555738243493886949097787,
3.65356790230978679260145290711, 4.46770805485337710679484201296, 6.91724796813801498388731951059, 9.784582996630399379667134259346, 10.90801870217700156786753403229, 12.07556339270585432864221002163, 13.35155752366317473699415555350, 14.63134521542414897201145037180, 15.70744023126281134172170356554, 16.87485248784751118223030738614
