L(s) = 1 | + (−27.8 + 27.8i)2-s + (−131. + 48.2i)3-s − 1.03e3i·4-s + (2.32e3 − 5.00e3i)6-s + (6.30e3 + 6.30e3i)7-s + (1.45e4 + 1.45e4i)8-s + (1.50e4 − 1.27e4i)9-s + 1.00e4i·11-s + (5.00e4 + 1.36e5i)12-s + (−1.13e5 + 1.13e5i)13-s − 3.50e5·14-s − 2.80e5·16-s + (4.18e5 − 4.18e5i)17-s + (−6.41e4 + 7.71e5i)18-s + 7.43e4i·19-s + ⋯ |
L(s) = 1 | + (−1.22 + 1.22i)2-s + (−0.938 + 0.344i)3-s − 2.02i·4-s + (0.731 − 1.57i)6-s + (0.991 + 0.991i)7-s + (1.25 + 1.25i)8-s + (0.763 − 0.646i)9-s + 0.206i·11-s + (0.696 + 1.89i)12-s + (−1.10 + 1.10i)13-s − 2.43·14-s − 1.07·16-s + (1.21 − 1.21i)17-s + (−0.144 + 1.73i)18-s + 0.130i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.255548 - 0.0259435i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.255548 - 0.0259435i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (131. - 48.2i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (27.8 - 27.8i)T - 512iT^{2} \) |
| 7 | \( 1 + (-6.30e3 - 6.30e3i)T + 4.03e7iT^{2} \) |
| 11 | \( 1 - 1.00e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + (1.13e5 - 1.13e5i)T - 1.06e10iT^{2} \) |
| 17 | \( 1 + (-4.18e5 + 4.18e5i)T - 1.18e11iT^{2} \) |
| 19 | \( 1 - 7.43e4iT - 3.22e11T^{2} \) |
| 23 | \( 1 + (1.15e6 + 1.15e6i)T + 1.80e12iT^{2} \) |
| 29 | \( 1 + 5.61e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.72e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + (-5.75e6 - 5.75e6i)T + 1.29e14iT^{2} \) |
| 41 | \( 1 + 1.54e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + (2.43e7 - 2.43e7i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 + (8.82e6 - 8.82e6i)T - 1.11e15iT^{2} \) |
| 53 | \( 1 + (4.01e7 + 4.01e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 - 1.04e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.51e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + (5.07e7 + 5.07e7i)T + 2.72e16iT^{2} \) |
| 71 | \( 1 - 4.26e7iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (-3.98e7 + 3.98e7i)T - 5.88e16iT^{2} \) |
| 79 | \( 1 + 1.98e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 + (-1.82e7 - 1.82e7i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 - 7.56e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (8.05e8 + 8.05e8i)T + 7.60e17iT^{2} \) |
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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
−12.18923112563841298007774267591, −11.39667797890508446271089941283, −9.928926802268576897084654880958, −9.280685279191258862181268836120, −7.926803576888979888887823498027, −6.90657326386774839877528629519, −5.64984173012284654249708046945, −4.82638485803227785993640959500, −1.74519845206968323086732713241, −0.16420147883217560514570593755,
0.941008055204064641353037824142, 1.87167136854507473916630458684, 3.75565583071546426657663485729, 5.43790114306611053328217975232, 7.53845928693069929073574286392, 7.977605827231769560078135161456, 9.866960674679816570751119042630, 10.51801511118201419796461557021, 11.34416166592017599679866640308, 12.25735909175553468485832837994