L(s) = 1 | + (4.37 + 2.52i)2-s + (−5.16 + 0.559i)3-s + (8.77 + 15.1i)4-s + (10.8 + 2.71i)5-s + (−24.0 − 10.6i)6-s + (−18.1 − 10.5i)7-s + 48.2i·8-s + (26.3 − 5.78i)9-s + (40.6 + 39.2i)10-s + (14.2 − 24.7i)11-s + (−53.8 − 73.5i)12-s + (8.72 − 5.03i)13-s + (−53.1 − 91.9i)14-s + (−57.5 − 7.93i)15-s + (−51.7 + 89.6i)16-s − 82.7i·17-s + ⋯ |
L(s) = 1 | + (1.54 + 0.893i)2-s + (−0.994 + 0.107i)3-s + (1.09 + 1.89i)4-s + (0.970 + 0.242i)5-s + (−1.63 − 0.721i)6-s + (−0.982 − 0.567i)7-s + 2.13i·8-s + (0.976 − 0.214i)9-s + (1.28 + 1.24i)10-s + (0.391 − 0.678i)11-s + (−1.29 − 1.77i)12-s + (0.186 − 0.107i)13-s + (−1.01 − 1.75i)14-s + (−0.990 − 0.136i)15-s + (−0.808 + 1.40i)16-s − 1.18i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.80462 + 1.46994i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.80462 + 1.46994i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (5.16 - 0.559i)T \) |
| 5 | \( 1 + (-10.8 - 2.71i)T \) |
good | 2 | \( 1 + (-4.37 - 2.52i)T + (4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (18.1 + 10.5i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-14.2 + 24.7i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-8.72 + 5.03i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + 82.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 1.91T + 6.85e3T^{2} \) |
| 23 | \( 1 + (147. - 85.2i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (128. - 222. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (24.0 + 41.6i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 161. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (139. + 241. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-233. - 134. i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-8.30 - 4.79i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 35.7iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (281. + 487. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-39.6 + 68.6i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-404. + 233. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 316.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 633. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (395. - 685. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-197. - 114. i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 53.9T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-83.7 - 48.3i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−15.73352822213414571452382539653, −14.18151555564438431727697878536, −13.46044840317340609437208635547, −12.51348505968389043661239782754, −11.21153210058086752339917294110, −9.713916089858742956545940312473, −7.12811311885383503693896938053, −6.25745776135149598544050941915, −5.30945574327203052722357706220, −3.57633455957613299228725377433,
1.99923764033196621490645994342, 4.21877259735353131480783391862, 5.78554569855625078870649885268, 6.35235862864338145230553594908, 9.684308305144139671534768689897, 10.58472968303159703446062770232, 12.02844840721151107079885213576, 12.66276310278491149011400222704, 13.45989992408266307912570791822, 14.81297864500951198860883585695