L(s) = 1 | + (−0.707 + 0.707i)2-s + (1.12 − 1.31i)3-s − 1.00i·4-s + (−0.192 + 2.22i)5-s + (0.133 + 1.72i)6-s + (1.96 + 1.96i)7-s + (0.707 + 0.707i)8-s + (−0.461 − 2.96i)9-s + (−1.43 − 1.71i)10-s + 2.97i·11-s + (−1.31 − 1.12i)12-s + (−1.33 + 1.33i)13-s − 2.77·14-s + (2.71 + 2.76i)15-s − 1.00·16-s + (0.217 − 0.217i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.650 − 0.759i)3-s − 0.500i·4-s + (−0.0860 + 0.996i)5-s + (0.0545 + 0.704i)6-s + (0.740 + 0.740i)7-s + (0.250 + 0.250i)8-s + (−0.153 − 0.988i)9-s + (−0.455 − 0.541i)10-s + 0.896i·11-s + (−0.379 − 0.325i)12-s + (−0.369 + 0.369i)13-s − 0.740·14-s + (0.700 + 0.713i)15-s − 0.250·16-s + (0.0527 − 0.0527i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11392 + 0.873514i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11392 + 0.873514i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.12 + 1.31i)T \) |
| 5 | \( 1 + (0.192 - 2.22i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-1.96 - 1.96i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.97iT - 11T^{2} \) |
| 13 | \( 1 + (1.33 - 1.33i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.217 + 0.217i)T - 17iT^{2} \) |
| 19 | \( 1 - 6.42iT - 19T^{2} \) |
| 29 | \( 1 + 1.22T + 29T^{2} \) |
| 31 | \( 1 - 6.33T + 31T^{2} \) |
| 37 | \( 1 + (-1.43 - 1.43i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.04iT - 41T^{2} \) |
| 43 | \( 1 + (1.80 - 1.80i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.199 + 0.199i)T - 47iT^{2} \) |
| 53 | \( 1 + (-9.17 - 9.17i)T + 53iT^{2} \) |
| 59 | \( 1 - 9.43T + 59T^{2} \) |
| 61 | \( 1 - 5.57T + 61T^{2} \) |
| 67 | \( 1 + (-9.41 - 9.41i)T + 67iT^{2} \) |
| 71 | \( 1 + 7.64iT - 71T^{2} \) |
| 73 | \( 1 + (4.05 - 4.05i)T - 73iT^{2} \) |
| 79 | \( 1 + 14.4iT - 79T^{2} \) |
| 83 | \( 1 + (12.1 + 12.1i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.45T + 89T^{2} \) |
| 97 | \( 1 + (-8.98 - 8.98i)T + 97iT^{2} \) |
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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
−10.36200684385697684825866000649, −9.732704178279808488545266709473, −8.688275041827222897371533422585, −7.954859797619476805667238148077, −7.29372335806844847473953599549, −6.49876712265175828352913734389, −5.56876817291574446676879614033, −4.09793981528967394637866691174, −2.60156116558521086980189893549, −1.74455279842357543115424733236,
0.854276851751856372026170134688, 2.43571767334316011866763236296, 3.70538009464117043831560651896, 4.58228913159324932196310550497, 5.38220049317170786354101667773, 7.13832855383238102484937561071, 8.226351713551377798987344517858, 8.462357002220744278939164563733, 9.476133338969256663488531264945, 10.13809064981122559070090802091