| L(s) = 1 | + (1.36 − 0.354i)2-s + (0.378 − 2.50i)3-s + (1.74 − 0.971i)4-s + (1.47 − 0.580i)5-s + (−0.372 − 3.56i)6-s + (−2.45 + 0.977i)7-s + (2.04 − 1.95i)8-s + (−3.28 − 1.01i)9-s + (1.81 − 1.31i)10-s + (1.04 + 3.38i)11-s + (−1.77 − 4.75i)12-s + (−2.45 − 0.561i)13-s + (−3.01 + 2.21i)14-s + (−0.896 − 3.92i)15-s + (2.11 − 3.39i)16-s + (4.62 + 3.15i)17-s + ⋯ |
| L(s) = 1 | + (0.967 − 0.251i)2-s + (0.218 − 1.44i)3-s + (0.873 − 0.485i)4-s + (0.661 − 0.259i)5-s + (−0.152 − 1.45i)6-s + (−0.929 + 0.369i)7-s + (0.724 − 0.689i)8-s + (−1.09 − 0.337i)9-s + (0.574 − 0.417i)10-s + (0.315 + 1.02i)11-s + (−0.512 − 1.37i)12-s + (−0.681 − 0.155i)13-s + (−0.806 + 0.590i)14-s + (−0.231 − 1.01i)15-s + (0.527 − 0.849i)16-s + (1.12 + 0.765i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0956 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0956 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.76619 - 1.94401i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76619 - 1.94401i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 + 0.354i)T \) |
| 7 | \( 1 + (2.45 - 0.977i)T \) |
| good | 3 | \( 1 + (-0.378 + 2.50i)T + (-2.86 - 0.884i)T^{2} \) |
| 5 | \( 1 + (-1.47 + 0.580i)T + (3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (-1.04 - 3.38i)T + (-9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (2.45 + 0.561i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-4.62 - 3.15i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (-1.29 - 0.746i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.00 - 4.77i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (-0.882 - 1.83i)T + (-18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (4.73 + 8.20i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.56 - 0.492i)T + (36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (-3.66 + 4.59i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (0.520 - 0.415i)T + (9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (-1.14 - 1.06i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (5.33 - 0.399i)T + (52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (4.93 + 1.93i)T + (43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (-11.6 - 0.875i)T + (60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (0.113 - 0.0657i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.61 + 1.26i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (11.9 - 11.0i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (6.19 - 10.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.86 + 1.33i)T + (74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-10.3 - 3.19i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−11.61129991927956220906422817189, −9.967856195881801075828326636790, −9.590649540978766784279971074344, −7.81633306151332635530600799780, −7.20066996275840347935832597247, −6.07976840042043219676213381251, −5.60764222981453650439186361076, −3.87284117407338096427752000483, −2.46127558034480351842700048778, −1.58838614106480375459109232487,
2.77244135227625530937723396968, 3.56383218078167661985917469317, 4.57487007604418968533393755476, 5.67479984011243041145232871963, 6.43598661946149498457777375387, 7.70417884858977742619758014992, 9.041809171339824901385289809423, 9.981978270828879551484444061240, 10.42584875565952155398164523471, 11.54747188678210113281845631010
