| L(s) = 1 | + (−6.05 − 1.62i)2-s + (21.6 + 80.8i)3-s + (−409. − 236. i)4-s + (1.39e3 + 125. i)5-s − 524. i·6-s + (−6.35e3 + 63.6i)7-s + (4.36e3 + 4.36e3i)8-s + (1.09e4 − 6.33e3i)9-s + (−8.22e3 − 3.02e3i)10-s + (1.57e4 − 2.72e4i)11-s + (1.02e4 − 3.82e4i)12-s + (−5.19e4 + 5.19e4i)13-s + (3.85e4 + 9.92e3i)14-s + (1.99e4 + 1.15e5i)15-s + (1.01e5 + 1.76e5i)16-s + (−1.45e4 + 3.89e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.267 − 0.0717i)2-s + (0.154 + 0.576i)3-s + (−0.799 − 0.461i)4-s + (0.995 + 0.0901i)5-s − 0.165i·6-s + (−0.999 + 0.0100i)7-s + (0.376 + 0.376i)8-s + (0.557 − 0.322i)9-s + (−0.260 − 0.0955i)10-s + (0.324 − 0.561i)11-s + (0.142 − 0.531i)12-s + (−0.504 + 0.504i)13-s + (0.268 + 0.0690i)14-s + (0.101 + 0.587i)15-s + (0.387 + 0.671i)16-s + (−0.0422 + 0.0113i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.25084 + 0.736018i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25084 + 0.736018i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.39e3 - 125. i)T \) |
| 7 | \( 1 + (6.35e3 - 63.6i)T \) |
| good | 2 | \( 1 + (6.05 + 1.62i)T + (443. + 256i)T^{2} \) |
| 3 | \( 1 + (-21.6 - 80.8i)T + (-1.70e4 + 9.84e3i)T^{2} \) |
| 11 | \( 1 + (-1.57e4 + 2.72e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (5.19e4 - 5.19e4i)T - 1.06e10iT^{2} \) |
| 17 | \( 1 + (1.45e4 - 3.89e3i)T + (1.02e11 - 5.92e10i)T^{2} \) |
| 19 | \( 1 + (-3.91e5 - 6.78e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (1.58e5 - 5.90e5i)T + (-1.55e12 - 9.00e11i)T^{2} \) |
| 29 | \( 1 - 7.28e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 + (-8.04e6 - 4.64e6i)T + (1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (3.86e6 + 1.03e6i)T + (1.12e14 + 6.49e13i)T^{2} \) |
| 41 | \( 1 - 4.95e6iT - 3.27e14T^{2} \) |
| 43 | \( 1 + (9.30e6 + 9.30e6i)T + 5.02e14iT^{2} \) |
| 47 | \( 1 + (-1.15e7 + 4.32e7i)T + (-9.69e14 - 5.59e14i)T^{2} \) |
| 53 | \( 1 + (7.99e7 - 2.14e7i)T + (2.85e15 - 1.64e15i)T^{2} \) |
| 59 | \( 1 + (-5.95e7 + 1.03e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (8.24e7 - 4.75e7i)T + (5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-3.52e7 - 1.31e8i)T + (-2.35e16 + 1.36e16i)T^{2} \) |
| 71 | \( 1 - 2.95e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-4.67e7 - 1.74e8i)T + (-5.09e16 + 2.94e16i)T^{2} \) |
| 79 | \( 1 + (3.03e7 - 1.75e7i)T + (5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-2.43e8 + 2.43e8i)T - 1.86e17iT^{2} \) |
| 89 | \( 1 + (3.20e7 + 5.54e7i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-9.43e8 - 9.43e8i)T + 7.60e17iT^{2} \) |
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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
−14.50366106309083179403744430436, −13.73226501772692207975958392122, −12.47401861172058724723852168473, −10.36472402403938011125177779410, −9.737972881327488442103383811122, −8.880891487087139570115561360574, −6.60222716326240307429592132131, −5.18473891716064710284087426512, −3.50174226278910780219386566564, −1.28880342964991497515609529538,
0.69311909742937523053756525893, 2.59534826209516815224144722056, 4.64038002286109224802390106642, 6.49700586586364926300014852057, 7.78979462978875501582051125495, 9.407608750049787222999946540060, 10.01532189639176545304233149491, 12.33687335376155111280368581250, 13.22312174233437805100759954238, 13.81342565121451274240792796586
