L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s + 1.00i·4-s + (−0.899 − 2.04i)5-s − 1.00·6-s + (−1.80 + 1.80i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (−0.811 + 2.08i)10-s + 4.16i·11-s + (0.707 + 0.707i)12-s + (−3.18 + 3.18i)13-s + 2.55·14-s + (−2.08 − 0.811i)15-s − 1.00·16-s + (−1.68 + 1.68i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.408 − 0.408i)3-s + 0.500i·4-s + (−0.402 − 0.915i)5-s − 0.408·6-s + (−0.683 + 0.683i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.256 + 0.658i)10-s + 1.25i·11-s + (0.204 + 0.204i)12-s + (−0.884 + 0.884i)13-s + 0.683·14-s + (−0.537 − 0.209i)15-s − 0.250·16-s + (−0.408 + 0.408i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.357589 + 0.299391i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.357589 + 0.299391i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.899 + 2.04i)T \) |
| 23 | \( 1 + (-4.36 + 1.98i)T \) |
good | 7 | \( 1 + (1.80 - 1.80i)T - 7iT^{2} \) |
| 11 | \( 1 - 4.16iT - 11T^{2} \) |
| 13 | \( 1 + (3.18 - 3.18i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.68 - 1.68i)T - 17iT^{2} \) |
| 19 | \( 1 + 5.24T + 19T^{2} \) |
| 29 | \( 1 - 3.48iT - 29T^{2} \) |
| 31 | \( 1 - 7.27T + 31T^{2} \) |
| 37 | \( 1 + (5.46 - 5.46i)T - 37iT^{2} \) |
| 41 | \( 1 - 4.53T + 41T^{2} \) |
| 43 | \( 1 + (6.37 + 6.37i)T + 43iT^{2} \) |
| 47 | \( 1 + (9.09 + 9.09i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.56 - 6.56i)T + 53iT^{2} \) |
| 59 | \( 1 - 6.80iT - 59T^{2} \) |
| 61 | \( 1 + 0.759iT - 61T^{2} \) |
| 67 | \( 1 + (2.02 - 2.02i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.56T + 71T^{2} \) |
| 73 | \( 1 + (5.78 - 5.78i)T - 73iT^{2} \) |
| 79 | \( 1 - 6.84T + 79T^{2} \) |
| 83 | \( 1 + (-2.40 - 2.40i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.58T + 89T^{2} \) |
| 97 | \( 1 + (4.03 - 4.03i)T - 97iT^{2} \) |
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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
−10.45543036244082258426570790075, −9.642095127199364620121721437289, −8.879387189814544605263217294974, −8.426902743299365575260336737408, −7.21021737566927101601692984791, −6.56986515511027892944768733694, −4.95121588500881153765770889778, −4.12971922469913296182172363964, −2.67265193543833931389562072553, −1.70270361280158901435249662199,
0.26343629758360899445946587784, 2.69124707992200514009737522965, 3.51941790743703101900199165918, 4.75556300631017493447590706629, 6.09682146648883431217617461181, 6.83029285271762859758326312833, 7.73297096791431156705481484367, 8.421791015595368442452561709211, 9.476375172917604058965118451820, 10.25794444560127499328778739531