L(s) = 1 | + 0.136·2-s − 2.96·3-s − 1.98·4-s + 3.29·5-s − 0.404·6-s − 0.543·8-s + 5.80·9-s + 0.448·10-s + 0.874·11-s + 5.87·12-s − 9.76·15-s + 3.88·16-s − 5.43·17-s + 0.791·18-s + 1.86·19-s − 6.51·20-s + 0.119·22-s − 6.86·23-s + 1.61·24-s + 5.82·25-s − 8.31·27-s − 4.15·29-s − 1.33·30-s + 9.04·31-s + 1.61·32-s − 2.59·33-s − 0.742·34-s + ⋯ |
L(s) = 1 | + 0.0964·2-s − 1.71·3-s − 0.990·4-s + 1.47·5-s − 0.165·6-s − 0.192·8-s + 1.93·9-s + 0.141·10-s + 0.263·11-s + 1.69·12-s − 2.52·15-s + 0.972·16-s − 1.31·17-s + 0.186·18-s + 0.428·19-s − 1.45·20-s + 0.0254·22-s − 1.43·23-s + 0.329·24-s + 1.16·25-s − 1.59·27-s − 0.771·29-s − 0.243·30-s + 1.62·31-s + 0.285·32-s − 0.451·33-s − 0.127·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9542388218\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9542388218\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.136T + 2T^{2} \) |
| 3 | \( 1 + 2.96T + 3T^{2} \) |
| 5 | \( 1 - 3.29T + 5T^{2} \) |
| 11 | \( 1 - 0.874T + 11T^{2} \) |
| 17 | \( 1 + 5.43T + 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 + 6.86T + 23T^{2} \) |
| 29 | \( 1 + 4.15T + 29T^{2} \) |
| 31 | \( 1 - 9.04T + 31T^{2} \) |
| 37 | \( 1 - 0.719T + 37T^{2} \) |
| 41 | \( 1 + 0.916T + 41T^{2} \) |
| 43 | \( 1 - 5.76T + 43T^{2} \) |
| 47 | \( 1 + 3.51T + 47T^{2} \) |
| 53 | \( 1 - 4.82T + 53T^{2} \) |
| 59 | \( 1 - 4.84T + 59T^{2} \) |
| 61 | \( 1 - 0.968T + 61T^{2} \) |
| 67 | \( 1 - 6.84T + 67T^{2} \) |
| 71 | \( 1 + 6.91T + 71T^{2} \) |
| 73 | \( 1 + 3.29T + 73T^{2} \) |
| 79 | \( 1 - 9.00T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 9.67T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−7.72034927042593410141305325219, −6.64520224032374371116044928546, −6.30239042393509416182991182080, −5.67204188220373127639370954081, −5.20129899117363291259553184328, −4.50654859844826803398372288619, −3.89600783845139331004155340570, −2.44653942107833407343438625171, −1.50250942092932138146382469041, −0.55104051717726143887529756823,
0.55104051717726143887529756823, 1.50250942092932138146382469041, 2.44653942107833407343438625171, 3.89600783845139331004155340570, 4.50654859844826803398372288619, 5.20129899117363291259553184328, 5.67204188220373127639370954081, 6.30239042393509416182991182080, 6.64520224032374371116044928546, 7.72034927042593410141305325219