L(s) = 1 | + (−0.707 + 0.707i)2-s + (1.53 + 1.53i)3-s − 1.00i·4-s + (−1.06 + 1.96i)5-s − 2.16·6-s + (−1.64 + 1.64i)7-s + (0.707 + 0.707i)8-s + 1.69i·9-s + (−0.633 − 2.14i)10-s + 3.84·11-s + (1.53 − 1.53i)12-s + (−2.55 + 2.55i)13-s − 2.32i·14-s + (−4.64 + 1.37i)15-s − 1.00·16-s + (−1.69 − 1.69i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.884 + 0.884i)3-s − 0.500i·4-s + (−0.477 + 0.878i)5-s − 0.884·6-s + (−0.622 + 0.622i)7-s + (0.250 + 0.250i)8-s + 0.566i·9-s + (−0.200 − 0.678i)10-s + 1.15·11-s + (0.442 − 0.442i)12-s + (−0.708 + 0.708i)13-s − 0.622i·14-s + (−1.20 + 0.354i)15-s − 0.250·16-s + (−0.410 − 0.410i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.202544 + 1.09061i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.202544 + 1.09061i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.06 - 1.96i)T \) |
| 43 | \( 1 + (6.13 - 2.32i)T \) |
good | 3 | \( 1 + (-1.53 - 1.53i)T + 3iT^{2} \) |
| 7 | \( 1 + (1.64 - 1.64i)T - 7iT^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 + (2.55 - 2.55i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.69 + 1.69i)T + 17iT^{2} \) |
| 19 | \( 1 + 1.66T + 19T^{2} \) |
| 23 | \( 1 + (4.75 - 4.75i)T - 23iT^{2} \) |
| 29 | \( 1 - 3.07T + 29T^{2} \) |
| 31 | \( 1 - 9.51T + 31T^{2} \) |
| 37 | \( 1 + (3.52 - 3.52i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 47 | \( 1 + (-2.14 - 2.14i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.79 + 6.79i)T - 53iT^{2} \) |
| 59 | \( 1 + 5.14iT - 59T^{2} \) |
| 61 | \( 1 - 7.55iT - 61T^{2} \) |
| 67 | \( 1 + (-2.61 - 2.61i)T + 67iT^{2} \) |
| 71 | \( 1 + 7.21iT - 71T^{2} \) |
| 73 | \( 1 + (-6.06 - 6.06i)T + 73iT^{2} \) |
| 79 | \( 1 - 15.6iT - 79T^{2} \) |
| 83 | \( 1 + (-12.4 + 12.4i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.54T + 89T^{2} \) |
| 97 | \( 1 + (-7.33 - 7.33i)T + 97iT^{2} \) |
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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
−11.54695664882567632632949445628, −10.15989022217980881544528565678, −9.696083140767972049232253022671, −8.930454448705125876847452809147, −8.106291861779106264656239055602, −6.86976119348084946165600450041, −6.27569546530170080472009967607, −4.57548575983222665949979813596, −3.58997176773218085048684735691, −2.44689159312612227663340994109,
0.75577738140451924224343121735, 2.17032280704782110768924115734, 3.52274206006880630000890147443, 4.53774080942102494684951804086, 6.42414916022625389628382867565, 7.30824613536273116076309725909, 8.256240204869527717996415851311, 8.708243934659189748511402202237, 9.766261832019862608925560644940, 10.60852437038574073388703872672