| L(s) = 1 | − i·2-s + 2i·3-s + 7·4-s + 2·6-s − 7i·7-s − 15i·8-s + 23·9-s − 8·11-s + 14i·12-s − 28i·13-s − 7·14-s + 41·16-s + 54i·17-s − 23i·18-s + 110·19-s + ⋯ |
| L(s) = 1 | − 0.353i·2-s + 0.384i·3-s + 0.875·4-s + 0.136·6-s − 0.377i·7-s − 0.662i·8-s + 0.851·9-s − 0.219·11-s + 0.336i·12-s − 0.597i·13-s − 0.133·14-s + 0.640·16-s + 0.770i·17-s − 0.301i·18-s + 1.32·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.20489 - 0.520504i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.20489 - 0.520504i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 7iT \) |
| good | 2 | \( 1 + iT - 8T^{2} \) |
| 3 | \( 1 - 2iT - 27T^{2} \) |
| 11 | \( 1 + 8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 28iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 54iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 110T + 6.85e3T^{2} \) |
| 23 | \( 1 + 48iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 110T + 2.43e4T^{2} \) |
| 31 | \( 1 - 12T + 2.97e4T^{2} \) |
| 37 | \( 1 + 246iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 182T + 6.89e4T^{2} \) |
| 43 | \( 1 + 128iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 324iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 162iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 810T + 2.05e5T^{2} \) |
| 61 | \( 1 + 488T + 2.26e5T^{2} \) |
| 67 | \( 1 - 244iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 768T + 3.57e5T^{2} \) |
| 73 | \( 1 - 702iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 440T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 730T + 7.04e5T^{2} \) |
| 97 | \( 1 - 294iT - 9.12e5T^{2} \) |
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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
−12.21699941332613919768587362547, −10.95369059912753197642364404769, −10.39413407446232173101589410236, −9.489806935875057032778564735686, −7.88382543117156500114639500010, −7.04025301917831269176074755221, −5.75833116149581168666937891226, −4.22741959090532436832821733988, −2.96140416030272704971455182803, −1.26694384366406525194663368952,
1.51361398231829685909117102130, 2.96440100112846591963876103018, 4.86911190424213694165637752337, 6.16535263141143387277789160513, 7.14333905031893225862557619598, 7.85961738441811759588903644808, 9.295522566499420371597716280328, 10.35016475923393553510367068291, 11.61855731501713864484576760085, 12.09746437454691625741704466031
