| L(s) = 1 | + (−1.39 − 0.237i)2-s + (1.98 − 1.66i)3-s + (1.88 + 0.662i)4-s + (−2.53 − 0.920i)5-s + (−3.16 + 1.85i)6-s + (1.43 + 0.829i)7-s + (−2.47 − 1.37i)8-s + (0.645 − 3.66i)9-s + (3.30 + 1.88i)10-s + (2.94 − 1.69i)11-s + (4.85 − 1.82i)12-s + (−2.17 + 2.59i)13-s + (−1.80 − 1.49i)14-s + (−6.55 + 2.38i)15-s + (3.12 + 2.50i)16-s + (1.03 + 5.88i)17-s + ⋯ |
| L(s) = 1 | + (−0.985 − 0.168i)2-s + (1.14 − 0.961i)3-s + (0.943 + 0.331i)4-s + (−1.13 − 0.411i)5-s + (−1.29 + 0.755i)6-s + (0.543 + 0.313i)7-s + (−0.874 − 0.485i)8-s + (0.215 − 1.22i)9-s + (1.04 + 0.596i)10-s + (0.886 − 0.511i)11-s + (1.40 − 0.527i)12-s + (−0.603 + 0.719i)13-s + (−0.482 − 0.400i)14-s + (−1.69 + 0.616i)15-s + (0.780 + 0.625i)16-s + (0.251 + 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.688291 - 0.383403i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.688291 - 0.383403i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.237i)T \) |
| 19 | \( 1 + (4.28 + 0.772i)T \) |
| good | 3 | \( 1 + (-1.98 + 1.66i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (2.53 + 0.920i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.43 - 0.829i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.94 + 1.69i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.17 - 2.59i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.03 - 5.88i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.01 - 5.52i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (2.92 + 0.516i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.91 + 5.04i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.798iT - 37T^{2} \) |
| 41 | \( 1 + (-0.904 - 1.07i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.235 + 0.645i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (6.47 + 1.14i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (2.91 + 7.99i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.838 + 4.75i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.56 + 1.29i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.76 + 10.0i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (2.34 + 0.855i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (8.03 - 6.73i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-6.98 + 5.86i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.78 - 3.91i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.99 - 9.52i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (8.40 - 1.48i)T + (91.1 - 33.1i)T^{2} \) |
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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
−14.62030825959304451897835371396, −13.05378832568793604442629666393, −12.03968920433218893783448539141, −11.26247907860573509230281243686, −9.363269400227729906291895202415, −8.369295471402728668906098216236, −7.902063143728312878944189690492, −6.63619218901805678967949042810, −3.72104758094225642515281941745, −1.80620868221924994388192742654,
2.93954801739861997659352642254, 4.49108457200289868971778633399, 7.07566851305740718613523670490, 8.001462112337258808023618340137, 8.978951172002281314456131747366, 10.05036210848522702657739877500, 11.03042207221379011637789671009, 12.17301052331099342578808208688, 14.39309850714952689949918971763, 14.80935369909792460545919982412
