| L(s) = 1 | + (0.994 − 0.108i)2-s + (−0.647 − 0.762i)3-s + (0.976 − 0.214i)4-s + (2.26 − 1.71i)5-s + (−0.725 − 0.687i)6-s + (0.732 + 1.83i)7-s + (0.947 − 0.319i)8-s + (−0.161 + 0.986i)9-s + (2.06 − 1.95i)10-s + (−0.851 − 0.394i)11-s + (−0.796 − 0.605i)12-s + (−0.231 − 1.40i)13-s + (0.927 + 1.74i)14-s + (−2.77 − 0.610i)15-s + (0.907 − 0.419i)16-s + (1.28 − 3.22i)17-s + ⋯ |
| L(s) = 1 | + (0.702 − 0.0764i)2-s + (−0.373 − 0.440i)3-s + (0.488 − 0.107i)4-s + (1.01 − 0.768i)5-s + (−0.296 − 0.280i)6-s + (0.276 + 0.694i)7-s + (0.335 − 0.112i)8-s + (−0.0539 + 0.328i)9-s + (0.652 − 0.617i)10-s + (−0.256 − 0.118i)11-s + (−0.229 − 0.174i)12-s + (−0.0640 − 0.390i)13-s + (0.247 + 0.467i)14-s + (−0.716 − 0.157i)15-s + (0.226 − 0.104i)16-s + (0.311 − 0.782i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.92828 - 0.792697i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.92828 - 0.792697i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.994 + 0.108i)T \) |
| 3 | \( 1 + (0.647 + 0.762i)T \) |
| 59 | \( 1 + (1.03 - 7.61i)T \) |
| good | 5 | \( 1 + (-2.26 + 1.71i)T + (1.33 - 4.81i)T^{2} \) |
| 7 | \( 1 + (-0.732 - 1.83i)T + (-5.08 + 4.81i)T^{2} \) |
| 11 | \( 1 + (0.851 + 0.394i)T + (7.12 + 8.38i)T^{2} \) |
| 13 | \( 1 + (0.231 + 1.40i)T + (-12.3 + 4.15i)T^{2} \) |
| 17 | \( 1 + (-1.28 + 3.22i)T + (-12.3 - 11.6i)T^{2} \) |
| 19 | \( 1 + (1.42 - 2.10i)T + (-7.03 - 17.6i)T^{2} \) |
| 23 | \( 1 + (0.00226 + 0.0418i)T + (-22.8 + 2.48i)T^{2} \) |
| 29 | \( 1 + (1.33 + 0.145i)T + (28.3 + 6.23i)T^{2} \) |
| 31 | \( 1 + (-0.111 - 0.164i)T + (-11.4 + 28.7i)T^{2} \) |
| 37 | \( 1 + (0.998 + 0.336i)T + (29.4 + 22.3i)T^{2} \) |
| 41 | \( 1 + (0.301 - 5.56i)T + (-40.7 - 4.43i)T^{2} \) |
| 43 | \( 1 + (9.59 - 4.43i)T + (27.8 - 32.7i)T^{2} \) |
| 47 | \( 1 + (-9.31 - 7.07i)T + (12.5 + 45.2i)T^{2} \) |
| 53 | \( 1 + (-0.306 - 0.290i)T + (2.86 + 52.9i)T^{2} \) |
| 61 | \( 1 + (13.6 - 1.48i)T + (59.5 - 13.1i)T^{2} \) |
| 67 | \( 1 + (6.67 - 2.24i)T + (53.3 - 40.5i)T^{2} \) |
| 71 | \( 1 + (1.51 + 1.15i)T + (18.9 + 68.4i)T^{2} \) |
| 73 | \( 1 + (-1.58 - 2.99i)T + (-40.9 + 60.4i)T^{2} \) |
| 79 | \( 1 + (5.50 - 6.47i)T + (-12.7 - 77.9i)T^{2} \) |
| 83 | \( 1 + (7.30 + 4.39i)T + (38.8 + 73.3i)T^{2} \) |
| 89 | \( 1 + (-9.79 - 1.06i)T + (86.9 + 19.1i)T^{2} \) |
| 97 | \( 1 + (4.74 - 8.95i)T + (-54.4 - 80.2i)T^{2} \) |
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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
−11.66351918842816093036789155100, −10.55583223161452457917472656949, −9.577584659115765642711198180705, −8.553874909054894002831629913053, −7.44434888773499647705133216352, −6.09532712779613737079430142141, −5.53271209574338597967529346784, −4.67530074240699397234980267019, −2.79793065190257347925351673153, −1.52991507682939462932543058229,
2.03357502504304731376418222802, 3.50155139407718881859900915642, 4.65465587202144873370147552726, 5.73189786057560329078909172964, 6.56800494037331330322233352885, 7.46950259450465836533779493464, 8.931151000143722705062335002133, 10.25308161339096405277234312427, 10.49971064311831516899846093081, 11.49505862544073064477101183580
