| L(s) = 1 | + (−1.30 + 1.25i)2-s + (0.0573 − 1.27i)4-s + (1.91 − 2.64i)5-s + (−0.412 − 0.0558i)7-s + (−0.858 − 0.982i)8-s + (0.793 + 5.85i)10-s + (0.629 − 0.376i)11-s + (−3.07 + 5.15i)13-s + (0.609 − 0.442i)14-s + (4.90 + 0.441i)16-s + (1.57 − 4.84i)17-s + (2.05 − 3.11i)19-s + (−3.26 − 2.60i)20-s + (−0.353 + 1.28i)22-s + (−1.29 − 0.624i)23-s + ⋯ |
| L(s) = 1 | + (−0.925 + 0.884i)2-s + (0.0286 − 0.638i)4-s + (0.857 − 1.18i)5-s + (−0.155 − 0.0211i)7-s + (−0.303 − 0.347i)8-s + (0.250 + 1.85i)10-s + (0.189 − 0.113i)11-s + (−0.853 + 1.42i)13-s + (0.162 − 0.118i)14-s + (1.22 + 0.110i)16-s + (0.382 − 1.17i)17-s + (0.471 − 0.713i)19-s + (−0.729 − 0.582i)20-s + (−0.0753 + 0.273i)22-s + (−0.270 − 0.130i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.859760 - 0.185702i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.859760 - 0.185702i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + (1.87 - 8.21i)T \) |
| good | 2 | \( 1 + (1.30 - 1.25i)T + (0.0897 - 1.99i)T^{2} \) |
| 5 | \( 1 + (-1.91 + 2.64i)T + (-1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (0.412 + 0.0558i)T + (6.74 + 1.86i)T^{2} \) |
| 11 | \( 1 + (-0.629 + 0.376i)T + (5.21 - 9.68i)T^{2} \) |
| 13 | \( 1 + (3.07 - 5.15i)T + (-6.16 - 11.4i)T^{2} \) |
| 17 | \( 1 + (-1.57 + 4.84i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.05 + 3.11i)T + (-7.46 - 17.4i)T^{2} \) |
| 23 | \( 1 + (1.29 + 0.624i)T + (14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (1.40 + 5.09i)T + (-24.8 + 14.8i)T^{2} \) |
| 31 | \( 1 + (0.903 + 10.0i)T + (-30.5 + 5.53i)T^{2} \) |
| 37 | \( 1 + (-1.28 + 0.619i)T + (23.0 - 28.9i)T^{2} \) |
| 41 | \( 1 + (-0.613 + 2.68i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-11.8 - 2.15i)T + (40.2 + 15.1i)T^{2} \) |
| 47 | \( 1 + (-1.96 + 0.735i)T + (35.3 - 30.9i)T^{2} \) |
| 53 | \( 1 + (0.154 + 3.43i)T + (-52.7 + 4.75i)T^{2} \) |
| 59 | \( 1 + (-1.16 + 2.72i)T + (-40.7 - 42.6i)T^{2} \) |
| 61 | \( 1 + (-8.61 + 1.16i)T + (58.8 - 16.2i)T^{2} \) |
| 67 | \( 1 + (7.09 + 0.318i)T + (66.7 + 6.00i)T^{2} \) |
| 73 | \( 1 + (0.921 + 0.963i)T + (-3.27 + 72.9i)T^{2} \) |
| 79 | \( 1 + (-8.42 + 7.35i)T + (10.6 - 78.2i)T^{2} \) |
| 83 | \( 1 + (-14.4 - 6.19i)T + (57.3 + 59.9i)T^{2} \) |
| 89 | \( 1 + (10.1 - 0.454i)T + (88.6 - 7.97i)T^{2} \) |
| 97 | \( 1 + (16.5 - 3.77i)T + (87.3 - 42.0i)T^{2} \) |
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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
−9.759323367779904189719177432801, −9.487810748275808800115256843042, −9.003470352575907908117572743321, −7.88785873865023642482011167667, −7.11931367348217210462955071636, −6.17050258435570569368530463156, −5.27029478159121476201309757312, −4.20183392463881790340680002399, −2.31202468854546131260157021142, −0.68011406649812139890462671066,
1.47586780121114218674746657504, 2.66796823465385716418108192534, 3.40564556657488239665269391936, 5.39950456403387610147370638458, 6.09634636239708344786686323333, 7.29581207584934261402197428437, 8.164813572963430863843517519575, 9.250120143874634360736888263943, 10.06304484699730112271854943864, 10.44494292900177837385776097626
