| L(s) = 1 | + (2.99 + 0.142i)2-s + (−0.301 + 1.02i)3-s + (−6.96 − 0.665i)4-s + (20.3 + 17.6i)5-s + (−1.05 + 3.03i)6-s + (20.4 + 39.6i)7-s + (−68.2 − 9.81i)8-s + (67.1 + 43.1i)9-s + (58.4 + 55.7i)10-s + (−9.39 + 3.25i)11-s + (2.78 − 6.95i)12-s + (121. + 154. i)13-s + (55.6 + 121. i)14-s + (−24.2 + 15.5i)15-s + (−93.3 − 17.9i)16-s + (69.6 − 6.64i)17-s + ⋯ |
| L(s) = 1 | + (0.749 + 0.0356i)2-s + (−0.0334 + 0.114i)3-s + (−0.435 − 0.0415i)4-s + (0.813 + 0.704i)5-s + (−0.0291 + 0.0842i)6-s + (0.417 + 0.809i)7-s + (−1.06 − 0.153i)8-s + (0.829 + 0.532i)9-s + (0.584 + 0.557i)10-s + (−0.0776 + 0.0268i)11-s + (0.0193 − 0.0482i)12-s + (0.720 + 0.915i)13-s + (0.283 + 0.621i)14-s + (−0.107 + 0.0691i)15-s + (−0.364 − 0.0702i)16-s + (0.240 − 0.0230i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.95084 + 1.19556i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.95084 + 1.19556i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 67 | \( 1 + (-4.47e3 + 411. i)T \) |
| good | 2 | \( 1 + (-2.99 - 0.142i)T + (15.9 + 1.52i)T^{2} \) |
| 3 | \( 1 + (0.301 - 1.02i)T + (-68.1 - 43.7i)T^{2} \) |
| 5 | \( 1 + (-20.3 - 17.6i)T + (88.9 + 618. i)T^{2} \) |
| 7 | \( 1 + (-20.4 - 39.6i)T + (-1.39e3 + 1.95e3i)T^{2} \) |
| 11 | \( 1 + (9.39 - 3.25i)T + (1.15e4 - 9.05e3i)T^{2} \) |
| 13 | \( 1 + (-121. - 154. i)T + (-6.73e3 + 2.77e4i)T^{2} \) |
| 17 | \( 1 + (-69.6 + 6.64i)T + (8.20e4 - 1.58e4i)T^{2} \) |
| 19 | \( 1 + (205. + 105. i)T + (7.55e4 + 1.06e5i)T^{2} \) |
| 23 | \( 1 + (117. + 483. i)T + (-2.48e5 + 1.28e5i)T^{2} \) |
| 29 | \( 1 + (349. - 605. i)T + (-3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-701. + 891. i)T + (-2.17e5 - 8.97e5i)T^{2} \) |
| 37 | \( 1 + (684. + 1.18e3i)T + (-9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (-989. + 704. i)T + (9.24e5 - 2.67e6i)T^{2} \) |
| 43 | \( 1 + (722. + 329. i)T + (2.23e6 + 2.58e6i)T^{2} \) |
| 47 | \( 1 + (-457. + 436. i)T + (2.32e5 - 4.87e6i)T^{2} \) |
| 53 | \( 1 + (950. - 434. i)T + (5.16e6 - 5.96e6i)T^{2} \) |
| 59 | \( 1 + (-134. + 935. i)T + (-1.16e7 - 3.41e6i)T^{2} \) |
| 61 | \( 1 + (-2.67e3 - 926. i)T + (1.08e7 + 8.55e6i)T^{2} \) |
| 71 | \( 1 + (2.14e3 + 205. i)T + (2.49e7 + 4.80e6i)T^{2} \) |
| 73 | \( 1 + (1.98e3 - 5.74e3i)T + (-2.23e7 - 1.75e7i)T^{2} \) |
| 79 | \( 1 + (2.53e3 - 6.32e3i)T + (-2.81e7 - 2.68e7i)T^{2} \) |
| 83 | \( 1 + (-1.11e4 - 2.15e3i)T + (4.40e7 + 1.76e7i)T^{2} \) |
| 89 | \( 1 + (-7.07e3 + 2.07e3i)T + (5.27e7 - 3.39e7i)T^{2} \) |
| 97 | \( 1 + (-1.10e4 + 6.35e3i)T + (4.42e7 - 7.66e7i)T^{2} \) |
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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
−14.18282178748996755309136045600, −13.35615606218015751964068288345, −12.30016415393321560300768445471, −10.91551140233878943764319686277, −9.715917127301825703648589452751, −8.573243706364306157921856908405, −6.65048112935003130931793356428, −5.50966357920006137474338286840, −4.18778708173443256032279235599, −2.24685690428196556601763750244,
1.14876046676140115651266678414, 3.71603032747058011297887975912, 4.98183384503221046553633312824, 6.18973679190161012310143268051, 7.996060806253245204067005206508, 9.323650972005377013081306102840, 10.38429951923862547188133161429, 12.04582935253217655390678022873, 13.17058811434016884708279307559, 13.47833387107057776391729304023
