| L(s) = 1 | + (−1.05 − 0.0919i)2-s + (−1.36 + 1.07i)3-s + (−0.873 − 0.154i)4-s + (1.52 − 1.00i)6-s + (−1.82 − 1.27i)7-s + (2.94 + 0.788i)8-s + (0.703 − 2.91i)9-s + (0.794 − 2.18i)11-s + (1.35 − 0.726i)12-s + (0.330 + 3.77i)13-s + (1.80 + 1.51i)14-s + (−1.35 − 0.491i)16-s + (−2.10 + 0.564i)17-s + (−1.00 + 2.99i)18-s + (5.74 − 3.31i)19-s + ⋯ |
| L(s) = 1 | + (−0.743 − 0.0650i)2-s + (−0.785 + 0.618i)3-s + (−0.436 − 0.0770i)4-s + (0.623 − 0.408i)6-s + (−0.690 − 0.483i)7-s + (1.04 + 0.278i)8-s + (0.234 − 0.972i)9-s + (0.239 − 0.657i)11-s + (0.390 − 0.209i)12-s + (0.0916 + 1.04i)13-s + (0.481 + 0.404i)14-s + (−0.337 − 0.122i)16-s + (−0.511 + 0.136i)17-s + (−0.237 + 0.707i)18-s + (1.31 − 0.760i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.155106 + 0.267039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.155106 + 0.267039i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.36 - 1.07i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (1.05 + 0.0919i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (1.82 + 1.27i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-0.794 + 2.18i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.330 - 3.77i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (2.10 - 0.564i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-5.74 + 3.31i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.791 + 1.13i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (3.09 - 2.59i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.911 - 5.17i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.60 + 5.98i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (7.40 - 8.82i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.72 - 3.70i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (4.87 - 6.95i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (2.80 - 2.80i)T - 53iT^{2} \) |
| 59 | \( 1 + (8.17 - 2.97i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.80 - 10.2i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-10.5 + 0.923i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-1.09 - 0.634i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.177 - 0.663i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-6.63 - 7.90i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.484 + 5.54i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-7.03 - 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (16.1 - 7.54i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.81467431040486094651026660439, −9.654944684339691302496829431947, −9.411873773387719149374481577526, −8.507936771991699495962316325110, −7.18724443859617564006739460758, −6.45024987125766341244916912352, −5.26529215864407735864145376573, −4.37846354197935882019367906629, −3.39458256863227466016848120712, −1.16736330344767361104491339656,
0.28258367036834946814043110916, 1.82336959725711133524576721225, 3.53969149435207399569414491774, 4.95252062176613002195044572950, 5.73911451478363697428015177070, 6.83252108037053432544430516830, 7.65074070310877297223766624658, 8.363653365229483456389202771913, 9.574593630074504053865727598442, 9.963157738869518772654956652879
