| L(s) = 1 | + (−0.571 − 1.75i)2-s + (−0.488 + 0.103i)3-s + (−1.15 + 0.836i)4-s + (−0.603 + 1.04i)5-s + (0.461 + 0.800i)6-s + (3.41 + 1.51i)7-s + (−0.863 − 0.627i)8-s + (−2.51 + 1.11i)9-s + (2.18 + 0.464i)10-s + (0.194 + 1.84i)11-s + (0.475 − 0.528i)12-s + (−3.46 − 3.85i)13-s + (0.721 − 6.86i)14-s + (0.186 − 0.573i)15-s + (−1.48 + 4.58i)16-s + (0.592 − 5.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.404 − 1.24i)2-s + (−0.282 + 0.0599i)3-s + (−0.575 + 0.418i)4-s + (−0.269 + 0.467i)5-s + (0.188 + 0.326i)6-s + (1.28 + 0.573i)7-s + (−0.305 − 0.221i)8-s + (−0.837 + 0.372i)9-s + (0.690 + 0.146i)10-s + (0.0585 + 0.556i)11-s + (0.137 − 0.152i)12-s + (−0.962 − 1.06i)13-s + (0.192 − 1.83i)14-s + (0.0481 − 0.148i)15-s + (−0.372 + 1.14i)16-s + (0.143 − 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.461135 - 0.327770i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.461135 - 0.327770i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (1.81 + 5.26i)T \) |
| good | 2 | \( 1 + (0.571 + 1.75i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (0.488 - 0.103i)T + (2.74 - 1.22i)T^{2} \) |
| 5 | \( 1 + (0.603 - 1.04i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-3.41 - 1.51i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.194 - 1.84i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (3.46 + 3.85i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.592 + 5.63i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (0.962 - 1.06i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-2.86 - 2.08i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.424 - 1.30i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (2.25 + 3.89i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.61 - 0.981i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (-4.38 + 4.87i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (1.30 - 4.02i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (11.8 - 5.27i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-2.13 + 0.453i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + 2.68T + 61T^{2} \) |
| 67 | \( 1 + (1.44 - 2.49i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.22 + 3.66i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (-0.439 - 4.18i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (1.17 - 11.1i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-10.2 - 2.17i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (-2.18 + 1.58i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-6.70 + 4.87i)T + (29.9 - 92.2i)T^{2} \) |
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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
−17.30145543880921557935682019049, −15.36587673522949301859522953738, −14.41043003583607054977161616318, −12.48658249053340011875077368244, −11.48166925227993287258754815304, −10.80332485836872513170851999697, −9.307519786002191128107663902690, −7.69038521508176830692022006880, −5.22379061394187740150474457877, −2.63129835239756389634579536525,
4.87106169569145169939996010039, 6.50114856772992223337955400878, 7.988921498470544467528313885202, 8.870270747005983952118131581241, 11.03433442204872852542808582688, 12.17007710885031703822306354357, 14.25543548812310783590304748979, 14.78964187204761275895953357358, 16.37980472836469985558065010034, 17.11681955971301396506816709424
