| L(s) = 1 | + (−12.0 − 1.91i)2-s + (10.2 + 14.1i)3-s + (81.8 + 26.5i)4-s + (10.5 + 3.43i)5-s + (−97.0 − 190. i)6-s + (−33.3 − 210. i)7-s + (−240. − 122. i)8-s + (131. − 403. i)9-s + (−121. − 61.8i)10-s + (−1.77e3 + 1.77e3i)11-s + (463. + 1.42e3i)12-s + 1.26e3·13-s + 2.60e3i·14-s + (59.9 + 184. i)15-s + (−1.77e3 − 1.29e3i)16-s + (210. + 412. i)17-s + ⋯ |
| L(s) = 1 | + (−1.51 − 0.239i)2-s + (0.379 + 0.522i)3-s + (1.27 + 0.415i)4-s + (0.0845 + 0.0274i)5-s + (−0.449 − 0.881i)6-s + (−0.0971 − 0.613i)7-s + (−0.470 − 0.239i)8-s + (0.180 − 0.554i)9-s + (−0.121 − 0.0618i)10-s + (−1.33 + 1.33i)11-s + (0.268 + 0.826i)12-s + 0.577·13-s + 0.950i·14-s + (0.0177 + 0.0546i)15-s + (−0.434 − 0.315i)16-s + (0.0428 + 0.0840i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.773181 - 0.312253i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.773181 - 0.312253i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 + (-2.09e5 + 8.72e4i)T \) |
| good | 2 | \( 1 + (12.0 + 1.91i)T + (60.8 + 19.7i)T^{2} \) |
| 3 | \( 1 + (-10.2 - 14.1i)T + (-225. + 693. i)T^{2} \) |
| 5 | \( 1 + (-10.5 - 3.43i)T + (1.26e4 + 9.18e3i)T^{2} \) |
| 7 | \( 1 + (33.3 + 210. i)T + (-1.11e5 + 3.63e4i)T^{2} \) |
| 11 | \( 1 + (1.77e3 - 1.77e3i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 - 1.26e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + (-210. - 412. i)T + (-1.41e7 + 1.95e7i)T^{2} \) |
| 19 | \( 1 + (1.49e3 + 2.05e3i)T + (-1.45e7 + 4.47e7i)T^{2} \) |
| 23 | \( 1 + (-1.88e4 + 9.61e3i)T + (8.70e7 - 1.19e8i)T^{2} \) |
| 29 | \( 1 + (-2.32e4 + 2.32e4i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 + (-2.05e4 + 3.24e3i)T + (8.44e8 - 2.74e8i)T^{2} \) |
| 37 | \( 1 + (3.14e4 - 4.97e3i)T + (2.44e9 - 7.92e8i)T^{2} \) |
| 41 | \( 1 + (-2.81e4 + 3.86e4i)T + (-1.46e9 - 4.51e9i)T^{2} \) |
| 43 | \( 1 + (-3.71e4 - 1.89e4i)T + (3.71e9 + 5.11e9i)T^{2} \) |
| 47 | \( 1 - 1.30e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (1.81e4 - 3.55e4i)T + (-1.30e10 - 1.79e10i)T^{2} \) |
| 59 | \( 1 + (3.79e3 - 2.39e4i)T + (-4.01e10 - 1.30e10i)T^{2} \) |
| 67 | \( 1 + (2.30e5 + 4.52e5i)T + (-5.31e10 + 7.31e10i)T^{2} \) |
| 71 | \( 1 + (-3.27e5 - 1.67e5i)T + (7.52e10 + 1.03e11i)T^{2} \) |
| 73 | \( 1 + (1.64e5 + 5.05e5i)T + (-1.22e11 + 8.89e10i)T^{2} \) |
| 79 | \( 1 + (-2.49e5 + 4.89e5i)T + (-1.42e11 - 1.96e11i)T^{2} \) |
| 83 | \( 1 + (6.78e5 - 4.92e5i)T + (1.01e11 - 3.10e11i)T^{2} \) |
| 89 | \( 1 + (1.11e5 - 7.05e5i)T + (-4.72e11 - 1.53e11i)T^{2} \) |
| 97 | \( 1 + (-1.51e5 + 2.08e5i)T + (-2.57e11 - 7.92e11i)T^{2} \) |
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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
−13.67327282784800686339735346434, −12.31896504654423402274456172216, −10.67938079034102062890800494191, −10.14154476776443857436884037324, −9.145846309196895152353466711782, −7.993936669071773887454949560207, −6.85124704829871112492560492727, −4.48910752607015369323378608575, −2.51289691883295342272127338792, −0.65817052818762481136255755909,
1.11765823188933196032262655487, 2.71926135952662354788702295459, 5.57650471884644278485241775756, 7.17667478363914391906377633153, 8.237570960017461657356590730868, 8.874620231912332312304214337740, 10.35354275132544270356910382733, 11.20019663752953924402805058417, 12.97522511310431802660016092152, 13.79289052440297167149668345423
