L(s) = 1 | − i·3-s + (2.23 − 0.0743i)5-s + 2.82i·7-s − 9-s − 6.46·11-s + i·13-s + (−0.0743 − 2.23i)15-s + 3.49i·17-s + 2.21·19-s + 2.82·21-s + 7.14i·23-s + (4.98 − 0.332i)25-s + i·27-s − 4.82·29-s − 9.65·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.999 − 0.0332i)5-s + 1.06i·7-s − 0.333·9-s − 1.95·11-s + 0.277i·13-s + (−0.0191 − 0.577i)15-s + 0.847i·17-s + 0.507·19-s + 0.617·21-s + 1.49i·23-s + (0.997 − 0.0664i)25-s + 0.192i·27-s − 0.896·29-s − 1.73·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0332 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0332 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.214095030\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.214095030\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2.23 + 0.0743i)T \) |
| 13 | \( 1 - iT \) |
good | 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 6.46T + 11T^{2} \) |
| 17 | \( 1 - 3.49iT - 17T^{2} \) |
| 19 | \( 1 - 2.21T + 19T^{2} \) |
| 23 | \( 1 - 7.14iT - 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 9.65T + 31T^{2} \) |
| 37 | \( 1 - 6.93iT - 37T^{2} \) |
| 41 | \( 1 + 0.148T + 41T^{2} \) |
| 43 | \( 1 + 3.03iT - 43T^{2} \) |
| 47 | \( 1 - 6.79iT - 47T^{2} \) |
| 53 | \( 1 + 1.70iT - 53T^{2} \) |
| 59 | \( 1 - 6.46T + 59T^{2} \) |
| 61 | \( 1 - 2.66T + 61T^{2} \) |
| 67 | \( 1 + 7.70iT - 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 5.76iT - 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 2.27iT - 83T^{2} \) |
| 89 | \( 1 - 9.21T + 89T^{2} \) |
| 97 | \( 1 + 8.11iT - 97T^{2} \) |
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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
−9.515929403888089945935758713177, −8.915488093266831323749003710341, −7.977782396483636045206026416943, −7.35594834609925223118855382252, −6.20637412147887014761255354386, −5.49263140406077109917206421401, −5.16780986400323261628063807146, −3.37300552935516608115193733045, −2.39140435046119567956236934495, −1.68517479021493962117793564076,
0.43335557391396205171008749605, 2.20788435781368937605566624201, 3.07260984292618042604436863267, 4.24393876829061025245587016028, 5.27955782105145920150578864203, 5.59911907864675778761732323322, 6.96554965401148065254067719497, 7.53872311023313026958267554001, 8.516206492921603424957241366998, 9.436886369193311339099402378386