L(s) = 1 | − i·2-s + (0.707 + 0.707i)3-s − 4-s + (1.45 − 1.69i)5-s + (0.707 − 0.707i)6-s + (−0.764 − 0.764i)7-s + i·8-s + 1.00i·9-s + (−1.69 − 1.45i)10-s + 4.86i·11-s + (−0.707 − 0.707i)12-s + 4.63i·13-s + (−0.764 + 0.764i)14-s + (2.22 − 0.170i)15-s + 16-s + 5.18·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (0.650 − 0.759i)5-s + (0.288 − 0.288i)6-s + (−0.288 − 0.288i)7-s + 0.353i·8-s + 0.333i·9-s + (−0.536 − 0.460i)10-s + 1.46i·11-s + (−0.204 − 0.204i)12-s + 1.28i·13-s + (−0.204 + 0.204i)14-s + (0.575 − 0.0441i)15-s + 0.250·16-s + 1.25·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.973668954\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.973668954\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1.45 + 1.69i)T \) |
| 37 | \( 1 + (-6.03 - 0.720i)T \) |
good | 7 | \( 1 + (0.764 + 0.764i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.86iT - 11T^{2} \) |
| 13 | \( 1 - 4.63iT - 13T^{2} \) |
| 17 | \( 1 - 5.18T + 17T^{2} \) |
| 19 | \( 1 + (-1.84 - 1.84i)T + 19iT^{2} \) |
| 23 | \( 1 - 4.72iT - 23T^{2} \) |
| 29 | \( 1 + (-6.46 + 6.46i)T - 29iT^{2} \) |
| 31 | \( 1 + (-1.27 - 1.27i)T + 31iT^{2} \) |
| 41 | \( 1 - 3.55iT - 41T^{2} \) |
| 43 | \( 1 + 7.49iT - 43T^{2} \) |
| 47 | \( 1 + (8.00 + 8.00i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.64 - 5.64i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.790 - 0.790i)T + 59iT^{2} \) |
| 61 | \( 1 + (5.24 + 5.24i)T + 61iT^{2} \) |
| 67 | \( 1 + (-0.777 + 0.777i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.599T + 71T^{2} \) |
| 73 | \( 1 + (-1.07 - 1.07i)T + 73iT^{2} \) |
| 79 | \( 1 + (-9.56 - 9.56i)T + 79iT^{2} \) |
| 83 | \( 1 + (11.2 - 11.2i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.70 + 1.70i)T - 89iT^{2} \) |
| 97 | \( 1 - 6.03T + 97T^{2} \) |
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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
−9.812787967499779462788819855224, −9.418070474046680090233733511611, −8.380513518728621799826814476539, −7.52259454960391635247344419576, −6.38953807654039115235169146367, −5.19918296551572773153590458897, −4.50359398676062240679957417071, −3.63561456347231234481501218944, −2.30121751573834526166964323485, −1.37440794059554517095115735639,
0.969981551396014842670380178087, 2.97889115555505894086193557552, 3.16472603087485516281901867369, 4.98527665207218283931540062607, 6.02369370746777568933416883199, 6.27039155377064439872191973784, 7.46091802300649510517992778773, 8.103468012330117537051001233872, 8.883375141169657597938094139761, 9.780017646032544096907406775539