L(s) = 1 | + (22.8 − 14.3i)3-s + (−93.9 + 82.4i)5-s − 279. i·7-s + (318. − 655. i)9-s − 772. i·11-s − 2.87e3i·13-s + (−968. + 3.23e3i)15-s + 2.79e3·17-s − 1.30e4·19-s + (−4.00e3 − 6.40e3i)21-s + 1.05e4·23-s + (2.02e3 − 1.54e4i)25-s + (−2.10e3 − 1.95e4i)27-s − 3.72e4i·29-s − 8.71e3·31-s + ⋯ |
L(s) = 1 | + (0.847 − 0.530i)3-s + (−0.751 + 0.659i)5-s − 0.815i·7-s + (0.436 − 0.899i)9-s − 0.580i·11-s − 1.30i·13-s + (−0.286 + 0.957i)15-s + 0.568·17-s − 1.90·19-s + (−0.432 − 0.691i)21-s + 0.867·23-s + (0.129 − 0.991i)25-s + (−0.107 − 0.994i)27-s − 1.52i·29-s − 0.292·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.01896 - 1.36882i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01896 - 1.36882i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-22.8 + 14.3i)T \) |
| 5 | \( 1 + (93.9 - 82.4i)T \) |
good | 7 | \( 1 + 279. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 772. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 2.87e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 2.79e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 1.30e4T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.05e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 3.72e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 8.71e3T + 8.87e8T^{2} \) |
| 37 | \( 1 + 3.55e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 1.08e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 6.66e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 4.02e3T + 1.07e10T^{2} \) |
| 53 | \( 1 - 7.95e4T + 2.21e10T^{2} \) |
| 59 | \( 1 - 1.51e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.07e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 3.63e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 3.79e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 4.61e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 2.27e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 9.98e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 1.03e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 7.34e5iT - 8.32e11T^{2} \) |
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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
−13.50691756940958679445086374074, −12.63202653504152006543610423348, −11.15555663388928933871868518108, −10.11768960767278263493138331017, −8.378631767951237154462022763704, −7.63396151974655965279100902987, −6.39349252578967442748428417014, −4.00998985948807334582395709799, −2.81964337680800346491339616358, −0.64135552506006148321730636308,
2.02779074027549781610676541851, 3.85177537198930116458037089091, 5.00624005020448084973055478232, 7.13006903136118081080862078988, 8.628927243895514125748142282758, 9.101753224012212991096065115736, 10.67307151470598575850732428227, 12.07089868389410379341117229988, 12.97281484749765220435857215306, 14.47859058851753727856079882174