L(s) = 1 | + (0.977 + 1.22i)3-s + (−0.470 + 0.374i)5-s + (−1.48 − 2.19i)7-s + (0.120 − 0.529i)9-s + (5.53 − 1.26i)11-s + (0.517 − 0.118i)13-s + (−0.918 − 0.209i)15-s + (0.00663 − 0.0137i)17-s + 4.54·19-s + (1.24 − 3.95i)21-s + (−0.102 − 0.212i)23-s + (−1.03 + 4.52i)25-s + (5.00 − 2.40i)27-s + (−3.07 − 1.48i)29-s + 5.84·31-s + ⋯ |
L(s) = 1 | + (0.564 + 0.707i)3-s + (−0.210 + 0.167i)5-s + (−0.559 − 0.828i)7-s + (0.0402 − 0.176i)9-s + (1.66 − 0.380i)11-s + (0.143 − 0.0327i)13-s + (−0.237 − 0.0541i)15-s + (0.00160 − 0.00334i)17-s + 1.04·19-s + (0.270 − 0.863i)21-s + (−0.0213 − 0.0443i)23-s + (−0.206 + 0.904i)25-s + (0.962 − 0.463i)27-s + (−0.571 − 0.275i)29-s + 1.05·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.88925 + 0.102495i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.88925 + 0.102495i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.48 + 2.19i)T \) |
good | 3 | \( 1 + (-0.977 - 1.22i)T + (-0.667 + 2.92i)T^{2} \) |
| 5 | \( 1 + (0.470 - 0.374i)T + (1.11 - 4.87i)T^{2} \) |
| 11 | \( 1 + (-5.53 + 1.26i)T + (9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.517 + 0.118i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.00663 + 0.0137i)T + (-10.5 - 13.2i)T^{2} \) |
| 19 | \( 1 - 4.54T + 19T^{2} \) |
| 23 | \( 1 + (0.102 + 0.212i)T + (-14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (3.07 + 1.48i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 - 5.84T + 31T^{2} \) |
| 37 | \( 1 + (-5.35 - 2.57i)T + (23.0 + 28.9i)T^{2} \) |
| 41 | \( 1 + (-1.23 + 0.983i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (5.47 + 4.36i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-0.699 - 3.06i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (10.7 - 5.16i)T + (33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 + (-6.17 + 7.74i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (3.48 - 7.23i)T + (-38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 + 5.76iT - 67T^{2} \) |
| 71 | \( 1 + (1.23 + 2.56i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-3.26 - 0.746i)T + (65.7 + 31.6i)T^{2} \) |
| 79 | \( 1 + 7.84iT - 79T^{2} \) |
| 83 | \( 1 + (3.22 - 14.1i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (4.67 + 1.06i)T + (80.1 + 38.6i)T^{2} \) |
| 97 | \( 1 + 13.2iT - 97T^{2} \) |
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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
−10.01605949496275600524905096624, −9.525499376702668485204163857352, −8.863138714173569309673324499325, −7.75224712217506337675946258757, −6.81489658001870893528895336800, −6.07317329172842621585451404058, −4.54955129955094791769730676068, −3.71242435475901799979186812428, −3.17975716535314754289477302765, −1.12751753805087504116879371477,
1.35911628958643222415183450710, 2.54483803843678322388177898755, 3.66347775286770202953178389977, 4.84665013146029990766754840419, 6.12580409417289266557362518537, 6.80190875989151127962958865819, 7.76033426834540893278804857950, 8.576639866264590114844987488589, 9.320348894379062440231286689212, 10.00424828778204144758863012336