L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.901 + 1.47i)3-s + 1.00i·4-s + (1.61 + 1.54i)5-s + (0.408 − 1.68i)6-s + (−1.95 + 1.95i)7-s + (0.707 − 0.707i)8-s + (−1.37 + 2.66i)9-s + (−0.0553 − 2.23i)10-s − 3.67i·11-s + (−1.47 + 0.901i)12-s + (2.60 + 2.60i)13-s + 2.76·14-s + (−0.819 + 3.78i)15-s − 1.00·16-s + (1.52 + 1.52i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.520 + 0.853i)3-s + 0.500i·4-s + (0.724 + 0.689i)5-s + (0.166 − 0.687i)6-s + (−0.737 + 0.737i)7-s + (0.250 − 0.250i)8-s + (−0.457 + 0.888i)9-s + (−0.0175 − 0.706i)10-s − 1.10i·11-s + (−0.426 + 0.260i)12-s + (0.722 + 0.722i)13-s + 0.737·14-s + (−0.211 + 0.977i)15-s − 0.250·16-s + (0.370 + 0.370i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0187 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0187 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.909623 + 0.926806i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.909623 + 0.926806i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.901 - 1.47i)T \) |
| 5 | \( 1 + (-1.61 - 1.54i)T \) |
| 19 | \( 1 - iT \) |
good | 7 | \( 1 + (1.95 - 1.95i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.67iT - 11T^{2} \) |
| 13 | \( 1 + (-2.60 - 2.60i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.52 - 1.52i)T + 17iT^{2} \) |
| 23 | \( 1 + (0.483 - 0.483i)T - 23iT^{2} \) |
| 29 | \( 1 + 8.02T + 29T^{2} \) |
| 31 | \( 1 + 3.06T + 31T^{2} \) |
| 37 | \( 1 + (-7.32 + 7.32i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.46iT - 41T^{2} \) |
| 43 | \( 1 + (3.11 + 3.11i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.44 - 3.44i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.51 - 7.51i)T - 53iT^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 0.989T + 61T^{2} \) |
| 67 | \( 1 + (-8.05 + 8.05i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.07iT - 71T^{2} \) |
| 73 | \( 1 + (-11.1 - 11.1i)T + 73iT^{2} \) |
| 79 | \( 1 + 7.15iT - 79T^{2} \) |
| 83 | \( 1 + (-2.66 + 2.66i)T - 83iT^{2} \) |
| 89 | \( 1 - 1.05T + 89T^{2} \) |
| 97 | \( 1 + (-5.51 + 5.51i)T - 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.02874821491135213341935172293, −9.902804932950261069219077623868, −9.323906017875790716899990798133, −8.741636765763707693196136807399, −7.66185760514249400368962284713, −6.24451004241576057844311477938, −5.60054166735395184242577391878, −3.84054735659685234750668684999, −3.13146935705677496652038187175, −2.05110467770869406644802820635,
0.828448373059475839957882867196, 2.12033339574763638004824218322, 3.67845050250215873123621179323, 5.19740186352321144348573815437, 6.21724337120432381454060212845, 7.01071754470196617035769929035, 7.80742592482794300520334336072, 8.694464251341614867246639176995, 9.629575620082598765492892554499, 10.02228239229843237372007897797