L(s) = 1 | + (0.766 + 0.642i)2-s + (0.479 − 0.402i)3-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)5-s + 0.625·6-s + (0.117 + 0.0425i)7-s + (−0.500 + 0.866i)8-s + (−0.452 + 2.56i)9-s + (0.5 + 0.866i)10-s + (1.69 − 2.93i)11-s + (0.479 + 0.402i)12-s + (1.17 + 6.64i)13-s + (0.0622 + 0.107i)14-s + (0.587 − 0.213i)15-s + (−0.939 + 0.342i)16-s + (0.825 − 4.67i)17-s + ⋯ |
L(s) = 1 | + (0.541 + 0.454i)2-s + (0.276 − 0.232i)3-s + (0.0868 + 0.492i)4-s + (0.420 + 0.152i)5-s + 0.255·6-s + (0.0442 + 0.0161i)7-s + (−0.176 + 0.306i)8-s + (−0.150 + 0.856i)9-s + (0.158 + 0.273i)10-s + (0.510 − 0.883i)11-s + (0.138 + 0.116i)12-s + (0.324 + 1.84i)13-s + (0.0166 + 0.0288i)14-s + (0.151 − 0.0552i)15-s + (−0.234 + 0.0855i)16-s + (0.200 − 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.90257 + 0.866382i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.90257 + 0.866382i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (3.50 + 4.97i)T \) |
good | 3 | \( 1 + (-0.479 + 0.402i)T + (0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-0.117 - 0.0425i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-1.69 + 2.93i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.17 - 6.64i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.825 + 4.67i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-3.58 + 3.00i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (1.45 + 2.51i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.53 - 2.66i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 8.39T + 31T^{2} \) |
| 41 | \( 1 + (0.863 + 4.89i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 - 5.19T + 43T^{2} \) |
| 47 | \( 1 + (-3.13 - 5.42i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.14 - 2.23i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-2.46 + 0.898i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.29 + 7.36i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.74 + 2.81i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-10.5 + 8.81i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + 4.64T + 73T^{2} \) |
| 79 | \( 1 + (10.5 + 3.83i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.17 - 6.65i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-5.00 + 1.81i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.55 - 2.69i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.45695760559826559975487702650, −10.95293393414859977405238902021, −9.315459761806815463994506176701, −8.846091940131272925328262460702, −7.50672440472132075945111582662, −6.81613600809814712332357112134, −5.71342058234684245495284859791, −4.70543926709530763413510953932, −3.36737601821121895083346367596, −2.00367311980458631344974388787,
1.47006438262449661078233072922, 3.15300784252019735805421881804, 3.99497468005192946348105497724, 5.44254048572864274251045854606, 6.12518974029742009443314180712, 7.52743276223821777556358407263, 8.640417432003151846219362382163, 9.767553462981758351750408447363, 10.18402901641887089332554614234, 11.34554608116737550392943934496