| L(s) = 1 | + (1.26 + 2.43i)2-s + (−0.0240 + 0.0675i)3-s + (−3.19 + 4.52i)4-s + (2.36 + 1.02i)5-s + (−0.195 + 0.0268i)6-s + (0.165 + 0.100i)7-s + (−9.64 − 1.32i)8-s + (2.32 + 1.89i)9-s + (0.483 + 7.07i)10-s + (2.57 − 0.720i)11-s + (−0.229 − 0.324i)12-s + (0.365 + 1.75i)13-s + (−0.0364 + 0.532i)14-s + (−0.126 + 0.135i)15-s + (−5.23 − 14.7i)16-s + (0.962 + 0.269i)17-s + ⋯ |
| L(s) = 1 | + (0.893 + 1.72i)2-s + (−0.0138 + 0.0390i)3-s + (−1.59 + 2.26i)4-s + (1.05 + 0.459i)5-s + (−0.0796 + 0.0109i)6-s + (0.0627 + 0.0381i)7-s + (−3.40 − 0.468i)8-s + (0.774 + 0.630i)9-s + (0.152 + 2.23i)10-s + (0.775 − 0.217i)11-s + (−0.0661 − 0.0937i)12-s + (0.101 + 0.487i)13-s + (−0.00973 + 0.142i)14-s + (−0.0326 + 0.0349i)15-s + (−1.30 − 3.68i)16-s + (0.233 + 0.0654i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.195312 - 2.65455i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.195312 - 2.65455i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-0.962 - 0.269i)T \) |
| 47 | \( 1 + (2.86 + 6.23i)T \) |
| good | 2 | \( 1 + (-1.26 - 2.43i)T + (-1.15 + 1.63i)T^{2} \) |
| 3 | \( 1 + (0.0240 - 0.0675i)T + (-2.32 - 1.89i)T^{2} \) |
| 5 | \( 1 + (-2.36 - 1.02i)T + (3.41 + 3.65i)T^{2} \) |
| 7 | \( 1 + (-0.165 - 0.100i)T + (3.22 + 6.21i)T^{2} \) |
| 11 | \( 1 + (-2.57 + 0.720i)T + (9.39 - 5.71i)T^{2} \) |
| 13 | \( 1 + (-0.365 - 1.75i)T + (-11.9 + 5.17i)T^{2} \) |
| 19 | \( 1 + (5.84 - 2.53i)T + (12.9 - 13.8i)T^{2} \) |
| 23 | \( 1 + (-3.54 + 6.83i)T + (-13.2 - 18.7i)T^{2} \) |
| 29 | \( 1 + (0.262 - 1.26i)T + (-26.5 - 11.5i)T^{2} \) |
| 31 | \( 1 + (3.32 + 9.35i)T + (-24.0 + 19.5i)T^{2} \) |
| 37 | \( 1 + (0.0640 + 0.936i)T + (-36.6 + 5.03i)T^{2} \) |
| 41 | \( 1 + (-9.87 + 1.35i)T + (39.4 - 11.0i)T^{2} \) |
| 43 | \( 1 + (6.29 - 8.92i)T + (-14.3 - 40.5i)T^{2} \) |
| 53 | \( 1 + (-5.97 + 0.821i)T + (51.0 - 14.2i)T^{2} \) |
| 59 | \( 1 + (-7.19 - 10.1i)T + (-19.7 + 55.5i)T^{2} \) |
| 61 | \( 1 + (-0.0634 + 0.927i)T + (-60.4 - 8.30i)T^{2} \) |
| 67 | \( 1 + (5.45 - 3.31i)T + (30.8 - 59.4i)T^{2} \) |
| 71 | \( 1 + (-4.07 + 7.85i)T + (-40.9 - 58.0i)T^{2} \) |
| 73 | \( 1 + (-1.16 + 0.947i)T + (14.8 - 71.4i)T^{2} \) |
| 79 | \( 1 + (-1.46 + 1.57i)T + (-5.39 - 78.8i)T^{2} \) |
| 83 | \( 1 + (2.97 - 0.834i)T + (70.9 - 43.1i)T^{2} \) |
| 89 | \( 1 + (6.12 + 2.66i)T + (60.7 + 65.0i)T^{2} \) |
| 97 | \( 1 + (3.17 - 8.93i)T + (-75.2 - 61.2i)T^{2} \) |
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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
−10.56522551771800588131469570822, −9.576616511482537472801302459172, −8.753990976209836756203100546834, −7.909394532602585320227334949509, −6.88628070153682094694128939680, −6.38383852971074335590942148554, −5.66672304536873367334094544330, −4.57602786196048316181449279108, −3.88353518137276195203955418013, −2.30779910495332981701806293907,
1.13770769064461216361704651170, 1.89407508086425161066732365673, 3.22445823537592287346752928272, 4.16444424503565495680010152404, 5.07695926540660848737864022810, 5.87602442413615187546282471038, 6.85743868692997072489137761524, 8.768888432385207175471336193138, 9.397536132656597576781566540833, 9.925116682804585933276089561834
