| L(s) = 1 | + (−0.0493 − 0.0678i)2-s + (−2.30 − 0.748i)3-s + (39.5 − 121. i)4-s + (53.9 + 274. i)5-s + (0.0627 + 0.193i)6-s − 911. i·7-s + (−20.4 + 6.63i)8-s + (−1.76e3 − 1.28e3i)9-s + (15.9 − 17.1i)10-s + (5.19e3 − 3.77e3i)11-s + (−182. + 250. i)12-s + (5.69e3 − 7.83e3i)13-s + (−61.8 + 44.9i)14-s + (81.0 − 671. i)15-s + (−1.32e4 − 9.62e3i)16-s + (−9.27e3 + 3.01e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.00435 − 0.00599i)2-s + (−0.0492 − 0.0160i)3-s + (0.308 − 0.951i)4-s + (0.192 + 0.981i)5-s + (0.000118 + 0.000365i)6-s − 1.00i·7-s + (−0.0141 + 0.00458i)8-s + (−0.806 − 0.586i)9-s + (0.00504 − 0.00543i)10-s + (1.17 − 0.855i)11-s + (−0.0304 + 0.0418i)12-s + (0.718 − 0.989i)13-s + (−0.00602 + 0.00437i)14-s + (0.00620 − 0.0514i)15-s + (−0.808 − 0.587i)16-s + (−0.458 + 0.148i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.27746 - 1.04486i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27746 - 1.04486i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-53.9 - 274. i)T \) |
| good | 2 | \( 1 + (0.0493 + 0.0678i)T + (-39.5 + 121. i)T^{2} \) |
| 3 | \( 1 + (2.30 + 0.748i)T + (1.76e3 + 1.28e3i)T^{2} \) |
| 7 | \( 1 + 911. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + (-5.19e3 + 3.77e3i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (-5.69e3 + 7.83e3i)T + (-1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (9.27e3 - 3.01e3i)T + (3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (7.34e3 + 2.25e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (-5.43e4 - 7.47e4i)T + (-1.05e9 + 3.23e9i)T^{2} \) |
| 29 | \( 1 + (3.24e4 - 9.98e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-5.35e4 - 1.64e5i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-2.94e5 + 4.04e5i)T + (-2.93e10 - 9.02e10i)T^{2} \) |
| 41 | \( 1 + (2.64e5 + 1.91e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 - 4.48e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + (-5.95e4 - 1.93e4i)T + (4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-1.46e6 - 4.75e5i)T + (9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (2.10e6 + 1.53e6i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (2.04e6 - 1.48e6i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (-3.55e6 + 1.15e6i)T + (4.90e12 - 3.56e12i)T^{2} \) |
| 71 | \( 1 + (3.87e5 - 1.19e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-1.72e6 - 2.36e6i)T + (-3.41e12 + 1.05e13i)T^{2} \) |
| 79 | \( 1 + (-8.55e5 + 2.63e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (7.99e5 - 2.59e5i)T + (2.19e13 - 1.59e13i)T^{2} \) |
| 89 | \( 1 + (-3.38e6 + 2.45e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (4.86e6 + 1.58e6i)T + (6.53e13 + 4.74e13i)T^{2} \) |
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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
−15.47205952486663923035022646472, −14.45298032836717422291495015784, −13.60500697998520848810561636254, −11.27160761868186619525657094378, −10.74823904938122251950388789456, −9.144074411480382103269986196190, −6.93935957281611528360960118328, −5.88720306889867634980368498205, −3.34437128424684944484857300091, −0.914135032570858297459129101588,
2.13123626372618978467728986565, 4.39846008681624539087227795004, 6.33218640826443045388069082955, 8.339436233785331053344484864940, 9.161884694850720684986957170474, 11.53217985832234617169658408084, 12.25097030397322461314529193861, 13.55028107206145788903442925052, 15.17651865388665373687158141585, 16.70026265633009429037275825968
