L(s) = 1 | + 2-s + 3-s + 4-s + 0.585·5-s + 6-s + 8-s + 9-s + 0.585·10-s + 5.93·11-s + 12-s + 6.45·13-s + 0.585·15-s + 16-s − 0.371·17-s + 18-s + 3.26·19-s + 0.585·20-s + 5.93·22-s + 23-s + 24-s − 4.65·25-s + 6.45·26-s + 27-s − 0.948·29-s + 0.585·30-s + 1.26·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.261·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.185·10-s + 1.78·11-s + 0.288·12-s + 1.79·13-s + 0.151·15-s + 0.250·16-s − 0.0901·17-s + 0.235·18-s + 0.747·19-s + 0.130·20-s + 1.26·22-s + 0.208·23-s + 0.204·24-s − 0.931·25-s + 1.26·26-s + 0.192·27-s − 0.176·29-s + 0.106·30-s + 0.226·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.565268023\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.565268023\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 11 | \( 1 - 5.93T + 11T^{2} \) |
| 13 | \( 1 - 6.45T + 13T^{2} \) |
| 17 | \( 1 + 0.371T + 17T^{2} \) |
| 19 | \( 1 - 3.26T + 19T^{2} \) |
| 29 | \( 1 + 0.948T + 29T^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 - 0.371T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 6.54T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 8.23T + 53T^{2} \) |
| 59 | \( 1 - 1.05T + 59T^{2} \) |
| 61 | \( 1 + 4.44T + 61T^{2} \) |
| 67 | \( 1 + 0.0286T + 67T^{2} \) |
| 71 | \( 1 - 2.38T + 71T^{2} \) |
| 73 | \( 1 - 4.76T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 2.51T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 8.85T + 97T^{2} \) |
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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
−8.004004790914866377383081454515, −7.09161220095080563302477279989, −6.46163648158524132527111341047, −5.99248073775134266337715682523, −5.11496613284156610403570399138, −4.05647369741984547701059810062, −3.73401829890525961926494350287, −2.99713995526637507318493328235, −1.73833058124905428566825319584, −1.23936912562065374082258074901,
1.23936912562065374082258074901, 1.73833058124905428566825319584, 2.99713995526637507318493328235, 3.73401829890525961926494350287, 4.05647369741984547701059810062, 5.11496613284156610403570399138, 5.99248073775134266337715682523, 6.46163648158524132527111341047, 7.09161220095080563302477279989, 8.004004790914866377383081454515