| L(s) = 1 | + (0.100 + 1.41i)2-s + (−5.03 − 1.09i)3-s + (−1.97 + 0.284i)4-s + (−4.91 + 0.920i)5-s + (1.03 − 7.20i)6-s + (−2.44 + 4.48i)7-s + (−0.601 − 2.76i)8-s + (15.9 + 7.27i)9-s + (−1.79 − 6.83i)10-s + (−14.1 − 16.2i)11-s + (10.2 + 0.734i)12-s + (3.78 + 6.92i)13-s + (−6.56 − 2.99i)14-s + (25.7 + 0.745i)15-s + (3.83 − 1.12i)16-s + (8.79 + 11.7i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 + 0.705i)2-s + (−1.67 − 0.364i)3-s + (−0.494 + 0.0711i)4-s + (−0.982 + 0.184i)5-s + (0.172 − 1.20i)6-s + (−0.349 + 0.640i)7-s + (−0.0751 − 0.345i)8-s + (1.77 + 0.808i)9-s + (−0.179 − 0.683i)10-s + (−1.28 − 1.48i)11-s + (0.856 + 0.0612i)12-s + (0.290 + 0.532i)13-s + (−0.469 − 0.214i)14-s + (1.71 + 0.0497i)15-s + (0.239 − 0.0704i)16-s + (0.517 + 0.691i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0600i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.476649 + 0.0143289i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.476649 + 0.0143289i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.100 - 1.41i)T \) |
| 5 | \( 1 + (4.91 - 0.920i)T \) |
| 23 | \( 1 + (-22.5 - 4.62i)T \) |
| good | 3 | \( 1 + (5.03 + 1.09i)T + (8.18 + 3.73i)T^{2} \) |
| 7 | \( 1 + (2.44 - 4.48i)T + (-26.4 - 41.2i)T^{2} \) |
| 11 | \( 1 + (14.1 + 16.2i)T + (-17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (-3.78 - 6.92i)T + (-91.3 + 142. i)T^{2} \) |
| 17 | \( 1 + (-8.79 - 11.7i)T + (-81.4 + 277. i)T^{2} \) |
| 19 | \( 1 + (-22.2 + 3.20i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (14.9 + 2.14i)T + (806. + 236. i)T^{2} \) |
| 31 | \( 1 + (12.2 + 7.88i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (3.26 + 8.75i)T + (-1.03e3 + 896. i)T^{2} \) |
| 41 | \( 1 + (19.3 + 42.4i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (60.3 + 13.1i)T + (1.68e3 + 768. i)T^{2} \) |
| 47 | \( 1 + (-32.8 + 32.8i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (13.9 + 7.61i)T + (1.51e3 + 2.36e3i)T^{2} \) |
| 59 | \( 1 + (-11.3 + 38.7i)T + (-2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (-70.7 - 45.4i)T + (1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (-6.14 - 85.9i)T + (-4.44e3 + 638. i)T^{2} \) |
| 71 | \( 1 + (-31.1 + 35.9i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (-1.52 + 2.03i)T + (-1.50e3 - 5.11e3i)T^{2} \) |
| 79 | \( 1 + (-17.9 + 61.0i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-108. + 40.3i)T + (5.20e3 - 4.51e3i)T^{2} \) |
| 89 | \( 1 + (-67.1 - 104. i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (-54.3 - 20.2i)T + (7.11e3 + 6.16e3i)T^{2} \) |
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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
−11.88453139514640949455857269227, −11.22242250769849543882106611147, −10.38846906050163814813455954995, −8.793237019966579202772270292383, −7.71410899887798161208700634561, −6.80756472428198687847281627346, −5.71098859623118880366487198399, −5.20174591523011464075099942498, −3.49170660095038229099974532965, −0.50666728430388649690681830580,
0.800316455461784261905473098413, 3.40269113902161780671234792758, 4.78012715617159850807706710093, 5.23039637910120565291893940076, 6.95360938018358179036552589169, 7.76832358334320772929879155499, 9.629704740678942251420034178348, 10.31621335946932993721047954630, 11.06148374860882822038520630705, 11.87157014846874134707989001688
