| L(s) = 1 | + (−1.41 + 0.0603i)2-s + (1.99 − 0.170i)4-s + (−0.0476 + 0.130i)5-s + (4.57 + 0.807i)7-s + (−2.80 + 0.361i)8-s + (0.0594 − 0.187i)10-s + (0.987 − 0.359i)11-s + (1.19 − 1.00i)13-s + (−6.51 − 0.864i)14-s + (3.94 − 0.680i)16-s + (5.25 − 3.03i)17-s + (−3.80 − 2.19i)19-s + (−0.0726 + 0.269i)20-s + (−1.37 + 0.567i)22-s + (0.440 + 2.49i)23-s + ⋯ |
| L(s) = 1 | + (−0.999 + 0.0427i)2-s + (0.996 − 0.0853i)4-s + (−0.0213 + 0.0585i)5-s + (1.73 + 0.305i)7-s + (−0.991 + 0.127i)8-s + (0.0187 − 0.0594i)10-s + (0.297 − 0.108i)11-s + (0.330 − 0.277i)13-s + (−1.74 − 0.230i)14-s + (0.985 − 0.170i)16-s + (1.27 − 0.736i)17-s + (−0.871 − 0.503i)19-s + (−0.0162 + 0.0601i)20-s + (−0.292 + 0.121i)22-s + (0.0917 + 0.520i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 972 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 972 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31869 - 0.00742905i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31869 - 0.00742905i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.0603i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.0476 - 0.130i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-4.57 - 0.807i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.987 + 0.359i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.19 + 1.00i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-5.25 + 3.03i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.80 + 2.19i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.440 - 2.49i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.395 + 0.471i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (4.88 - 0.861i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.43 + 2.49i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.23 + 5.04i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.748 + 2.05i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.91 - 10.8i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 8.73iT - 53T^{2} \) |
| 59 | \( 1 + (-7.71 - 2.80i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.94 + 11.0i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.62 - 4.32i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.33 - 4.04i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.64 - 9.77i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.07 + 1.28i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (1.79 + 1.50i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-7.63 - 4.40i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.29 + 1.92i)T + (74.3 - 62.3i)T^{2} \) |
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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
−9.966843667102226436724408592203, −8.988251357271620006625084770336, −8.427350026145385948431478558756, −7.65038692856746281744536586711, −6.95103839827613158673077044697, −5.67292204333003831248278343366, −5.00463077751840079517710287864, −3.47794562534061895782654233941, −2.15506851673393748901148754228, −1.11132242244199545246106090068,
1.19565478401415118198987799700, 2.04337653993838010091279323968, 3.62424845167381047144505095781, 4.75099917754694984527920144357, 5.86139136961013600068247686257, 6.84285824800690019637839102702, 7.80491360432474856916151326616, 8.341939386134019098593001159859, 8.945232168748149944613367926702, 10.28789783484525661354629449626
