L(s) = 1 | + (−1.5 + 2.59i)3-s + (−6.41 − 11.1i)5-s + (−4.5 − 7.79i)9-s + (18.4 − 31.8i)11-s − 87.1·13-s + 38.4·15-s + (51.2 − 88.8i)17-s + (47.9 + 82.9i)19-s + (48 + 83.1i)23-s + (−19.7 + 34.1i)25-s + 27·27-s − 212.·29-s + (−79.6 + 137. i)31-s + (55.2 + 95.6i)33-s + (−64.3 − 111. i)37-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.573 − 0.993i)5-s + (−0.166 − 0.288i)9-s + (0.504 − 0.874i)11-s − 1.85·13-s + 0.662·15-s + (0.731 − 1.26i)17-s + (0.578 + 1.00i)19-s + (0.435 + 0.753i)23-s + (−0.157 + 0.273i)25-s + 0.192·27-s − 1.35·29-s + (−0.461 + 0.799i)31-s + (0.291 + 0.504i)33-s + (−0.285 − 0.495i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3250180540\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3250180540\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 - 2.59i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (6.41 + 11.1i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-18.4 + 31.8i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 87.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-51.2 + 88.8i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-47.9 - 82.9i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-48 - 83.1i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 212.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (79.6 - 137. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (64.3 + 111. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 33.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-135. - 234. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (224. - 388. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (334. - 578. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (121. + 211. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-167. + 290. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 339.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (459. - 795. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-68.1 - 118. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 287.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-80.9 - 140. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 182.T + 9.12e5T^{2} \) |
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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
−10.60649222255342811830829067924, −9.401559698286666688504725892135, −9.229350801882202962062908787184, −7.84839847581291705424410934421, −7.27054161872513760086050775764, −5.67450165108154360326489512651, −5.10302214380054221955049077615, −4.09571843845541839398790917037, −2.99711963941344517688970088082, −1.10642008696984249009701560596,
0.11178613706943458676915876835, 1.92018403740098361058566443051, 3.02887206235046284375428809999, 4.29355258273341553393091532460, 5.38821474924641140247003884110, 6.63683367514686818768926234791, 7.28773823239609343454694544483, 7.80204387466211709911751275614, 9.240840324274242376986280801752, 10.05478354926338843230111530346