L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (0.500 + 0.866i)4-s − 2·5-s + (−0.499 − 0.866i)6-s + (−2 − 3.46i)7-s + 3·8-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)10-s + (2 − 3.46i)11-s + 12-s − 3.99·14-s + (−1 + 1.73i)15-s + (0.500 − 0.866i)16-s + (−1 − 1.73i)17-s − 0.999·18-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (0.250 + 0.433i)4-s − 0.894·5-s + (−0.204 − 0.353i)6-s + (−0.755 − 1.30i)7-s + 1.06·8-s + (−0.166 − 0.288i)9-s + (−0.316 + 0.547i)10-s + (0.603 − 1.04i)11-s + 0.288·12-s − 1.06·14-s + (−0.258 + 0.447i)15-s + (0.125 − 0.216i)16-s + (−0.242 − 0.420i)17-s − 0.235·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.776549 - 1.38576i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.776549 - 1.38576i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-6 - 10.3i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (1 - 1.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.91082762085108215318468509349, −9.929326569523744335718636515994, −8.675479495880407507416020894278, −7.72219644707522635571244564930, −7.16848144627570725278851980456, −6.19417251910214000081290352133, −4.23575502597630640115344810438, −3.73668906108316903959501848053, −2.73303667353826011515671348488, −0.839282436912720068322739163135,
2.12317333635998849505750016637, 3.58577020098926599084787960698, 4.65206823403212734468833470923, 5.61031594342196064957574046837, 6.60603110459454829388153230802, 7.41413062405619755433591484092, 8.584221031990018921512585595383, 9.356319336327605337714852632156, 10.25070766310604464622937920554, 11.20364562420934786870735072249