L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.992 − 0.832i)3-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)5-s − 1.29·6-s + (−3.09 − 1.12i)7-s + (0.500 − 0.866i)8-s + (−0.229 + 1.30i)9-s + (−0.5 − 0.866i)10-s + (3.04 − 5.28i)11-s + (0.992 + 0.832i)12-s + (−0.733 − 4.15i)13-s + (1.64 + 2.85i)14-s + (1.21 − 0.443i)15-s + (−0.939 + 0.342i)16-s + (0.443 − 2.51i)17-s + ⋯ |
L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.572 − 0.480i)3-s + (0.0868 + 0.492i)4-s + (0.420 + 0.152i)5-s − 0.528·6-s + (−1.17 − 0.426i)7-s + (0.176 − 0.306i)8-s + (−0.0764 + 0.433i)9-s + (−0.158 − 0.273i)10-s + (0.919 − 1.59i)11-s + (0.286 + 0.240i)12-s + (−0.203 − 1.15i)13-s + (0.440 + 0.763i)14-s + (0.314 − 0.114i)15-s + (−0.234 + 0.0855i)16-s + (0.107 − 0.609i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.773667 - 0.881523i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.773667 - 0.881523i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 5 | \( 1 + (-0.939 - 0.342i)T \) |
| 37 | \( 1 + (5.66 + 2.20i)T \) |
good | 3 | \( 1 + (-0.992 + 0.832i)T + (0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (3.09 + 1.12i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-3.04 + 5.28i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.733 + 4.15i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.443 + 2.51i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-4.21 + 3.53i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-0.278 - 0.482i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.76 - 6.52i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 41 | \( 1 + (1.13 + 6.42i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + 8.25T + 43T^{2} \) |
| 47 | \( 1 + (-3.46 - 6.00i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.81 - 0.659i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (9.75 - 3.55i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.19 - 12.4i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.69 - 3.16i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (10.3 - 8.72i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + (-12.4 - 4.51i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.17 + 6.64i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (0.306 - 0.111i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-3.19 - 5.53i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.98168124186787811273050163536, −10.21409599983343478580960257835, −9.251254550923781211072093435911, −8.556441201879930330749736452321, −7.45369478800065966216018655542, −6.63305644616305325416162402656, −5.39738758119179357363489319448, −3.36435438258621595237634664151, −2.86348689130910383929093663075, −0.942251281592240141920859257593,
1.90657681645787510358882065319, 3.50962358690131535871463129068, 4.69408568116600375109421191189, 6.29752268026272405370902842655, 6.69262676188924095354222768841, 8.086965633150003052108523695168, 9.223803367812780731862257064690, 9.668365379209894606354896003509, 10.01267599182577245488257676912, 11.85613489527185386266167189402