| L(s) = 1 | + (−4.50 − 2.16i)2-s + (4.54 − 5.69i)3-s + (10.5 + 13.2i)4-s + (−16.6 − 8.01i)5-s + (−32.8 + 15.8i)6-s + (−4.27 + 5.35i)7-s + (−9.98 − 43.7i)8-s + (−5.80 − 25.4i)9-s + (57.5 + 72.1i)10-s + (−0.169 + 0.743i)11-s + 123.·12-s + (15.7 − 69.1i)13-s + (30.8 − 14.8i)14-s + (−121. + 58.4i)15-s + (−19.6 + 86.2i)16-s + 65.5·17-s + ⋯ |
| L(s) = 1 | + (−1.59 − 0.766i)2-s + (0.874 − 1.09i)3-s + (1.32 + 1.65i)4-s + (−1.48 − 0.716i)5-s + (−2.23 + 1.07i)6-s + (−0.230 + 0.289i)7-s + (−0.441 − 1.93i)8-s + (−0.215 − 0.942i)9-s + (1.82 + 2.28i)10-s + (−0.00465 + 0.0203i)11-s + 2.97·12-s + (0.336 − 1.47i)13-s + (0.589 − 0.283i)14-s + (−2.08 + 1.00i)15-s + (−0.307 + 1.34i)16-s + 0.935·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0609147 - 0.525545i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0609147 - 0.525545i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (148. + 48.1i)T \) |
| good | 2 | \( 1 + (4.50 + 2.16i)T + (4.98 + 6.25i)T^{2} \) |
| 3 | \( 1 + (-4.54 + 5.69i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (16.6 + 8.01i)T + (77.9 + 97.7i)T^{2} \) |
| 7 | \( 1 + (4.27 - 5.35i)T + (-76.3 - 334. i)T^{2} \) |
| 11 | \( 1 + (0.169 - 0.743i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-15.7 + 69.1i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 - 65.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + (41.8 + 52.4i)T + (-1.52e3 + 6.68e3i)T^{2} \) |
| 23 | \( 1 + (-99.5 + 47.9i)T + (7.58e3 - 9.51e3i)T^{2} \) |
| 31 | \( 1 + (-66.9 - 32.2i)T + (1.85e4 + 2.32e4i)T^{2} \) |
| 37 | \( 1 + (18.4 + 80.7i)T + (-4.56e4 + 2.19e4i)T^{2} \) |
| 41 | \( 1 - 148.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-76.1 + 36.6i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (74.0 - 324. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (54.7 + 26.3i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 - 402.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (332. - 417. i)T + (-5.05e4 - 2.21e5i)T^{2} \) |
| 67 | \( 1 + (116. + 510. i)T + (-2.70e5 + 1.30e5i)T^{2} \) |
| 71 | \( 1 + (-14.9 + 65.5i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-210. + 101. i)T + (2.42e5 - 3.04e5i)T^{2} \) |
| 79 | \( 1 + (116. + 509. i)T + (-4.44e5 + 2.13e5i)T^{2} \) |
| 83 | \( 1 + (-32.3 - 40.5i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-762. - 366. i)T + (4.39e5 + 5.51e5i)T^{2} \) |
| 97 | \( 1 + (-537. - 673. i)T + (-2.03e5 + 8.89e5i)T^{2} \) |
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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
−16.36638058203805261656925499818, −15.17401309097340561825239130408, −12.88928850672421094569968227242, −12.26756306239184586970015609252, −10.90352427682963804621464112523, −9.033746283689833675211250858410, −8.121444879350661998932102632630, −7.47899431146211403148704245674, −3.04255603506188941756417864243, −0.793254040529847008489547422465,
3.77407850937366200642601345521, 6.93039393470561883598499881534, 8.081408302280736202006535181429, 9.173932807349759343544868077060, 10.32511551340031027795975998000, 11.44392210665245399544858505761, 14.46021323861732161681264603884, 15.14283968047420416991280046476, 16.08224416636431678070042971349, 16.75078071246364802545919630412
