| L(s) = 1 | + (2.06 + 1.50i)2-s + (−3.25 − 10.0i)3-s + (−7.86 − 24.2i)4-s + (−54.0 − 14.2i)5-s + (8.32 − 25.6i)6-s − 36.6·7-s + (45.4 − 139. i)8-s + (106. − 77.5i)9-s + (−90.4 − 110. i)10-s + (55.3 + 40.1i)11-s + (−217. + 157. i)12-s + (278. − 202. i)13-s + (−75.7 − 55.0i)14-s + (33.3 + 588. i)15-s + (−355. + 258. i)16-s + (−201. + 620. i)17-s + ⋯ |
| L(s) = 1 | + (0.365 + 0.265i)2-s + (−0.208 − 0.642i)3-s + (−0.245 − 0.756i)4-s + (−0.967 − 0.254i)5-s + (0.0944 − 0.290i)6-s − 0.282·7-s + (0.250 − 0.771i)8-s + (0.439 − 0.319i)9-s + (−0.285 − 0.350i)10-s + (0.137 + 0.100i)11-s + (−0.435 + 0.316i)12-s + (0.457 − 0.332i)13-s + (−0.103 − 0.0750i)14-s + (0.0382 + 0.674i)15-s + (−0.346 + 0.251i)16-s + (−0.169 + 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.688975 - 0.960741i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.688975 - 0.960741i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (54.0 + 14.2i)T \) |
| good | 2 | \( 1 + (-2.06 - 1.50i)T + (9.88 + 30.4i)T^{2} \) |
| 3 | \( 1 + (3.25 + 10.0i)T + (-196. + 142. i)T^{2} \) |
| 7 | \( 1 + 36.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-55.3 - 40.1i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-278. + 202. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (201. - 620. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-322. + 992. i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-1.65e3 - 1.20e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (1.24e3 + 3.81e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-2.94e3 + 9.04e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (3.17e3 - 2.30e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.18e4 + 8.57e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-7.13e3 - 2.19e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-7.92e3 - 2.43e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-7.62e3 + 5.54e3i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (2.54e4 + 1.84e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-1.43e3 + 4.40e3i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-1.01e4 - 3.13e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (6.69e4 + 4.86e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-2.16e4 - 6.66e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (5.53e3 - 1.70e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-1.14e5 - 8.33e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-4.06e3 - 1.24e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−15.76987292208663990018608678044, −15.13729892156887277652664428564, −13.48931571267720898623401857714, −12.55335616955182094176544770169, −11.09134734915985673612724374474, −9.374300969581697069685095768841, −7.50094887712389727254158817855, −6.10478746406103191797111433810, −4.20747811587513852367196165083, −0.796152578948690396687426849225,
3.42550037956250342089096902472, 4.68854851726911233507300346576, 7.25362862668355239952752840423, 8.777136184323510737609670741546, 10.63306537586759779346799048309, 11.76587722371867509078541631624, 12.94385778730780450905175449655, 14.39701893834793148761120070971, 15.94900234113097587890113860313, 16.50571999111660227889920167158
