| L(s) = 1 | + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (−1.18 + 2.59i)5-s + (−4.30 − 1.26i)7-s + (−0.841 + 0.540i)8-s + (−2.74 + 0.804i)10-s + (0.129 − 0.149i)11-s + (−3.55 + 1.04i)13-s + (−1.86 − 4.07i)14-s + (−0.959 − 0.281i)16-s + (−0.0880 − 0.612i)17-s + (0.0368 − 0.256i)19-s + (−2.40 − 1.54i)20-s + 0.198·22-s + (4.31 + 2.09i)23-s + ⋯ |
| L(s) = 1 | + (0.463 + 0.534i)2-s + (−0.0711 + 0.494i)4-s + (−0.530 + 1.16i)5-s + (−1.62 − 0.477i)7-s + (−0.297 + 0.191i)8-s + (−0.866 + 0.254i)10-s + (0.0391 − 0.0452i)11-s + (−0.986 + 0.289i)13-s + (−0.497 − 1.08i)14-s + (−0.239 − 0.0704i)16-s + (−0.0213 − 0.148i)17-s + (0.00846 − 0.0588i)19-s + (−0.537 − 0.345i)20-s + 0.0422·22-s + (0.899 + 0.436i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00446920 + 0.752750i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00446920 + 0.752750i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 - 0.755i)T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (-4.31 - 2.09i)T \) |
| good | 5 | \( 1 + (1.18 - 2.59i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (4.30 + 1.26i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-0.129 + 0.149i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (3.55 - 1.04i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.0880 + 0.612i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.0368 + 0.256i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.06 - 7.42i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (2.36 - 1.51i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-2.45 - 5.37i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (3.51 - 7.68i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (8.47 + 5.44i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 9.81T + 47T^{2} \) |
| 53 | \( 1 + (3.77 + 1.10i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (9.41 - 2.76i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (4.30 - 2.76i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-4.81 - 5.55i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-7.26 - 8.38i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-2.22 + 15.4i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-14.3 + 4.21i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (4.00 + 8.75i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (12.9 + 8.29i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-4.92 + 10.7i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.77280130779023822401160343231, −10.73436184982380038234284680633, −9.917115695287939406630110492871, −8.973514200111999509464137137005, −7.47052888047698029133723014753, −6.97315308167337570989262793485, −6.32359541030529009417979503262, −4.89117336372895551048644754295, −3.50056992614183806938765128235, −2.95816077100572030835814334160,
0.39728842314215496128054658846, 2.51436394474293581219119989142, 3.70193004761288402216903495980, 4.78870814467805642022739388745, 5.77255099133277709723342670192, 6.86061212270799774146545119447, 8.151170465418202439602603913210, 9.272712465898718449648605732816, 9.685760150125969477129295298227, 10.84756383523565880641781241416
