L(s) = 1 | − 5-s − 7-s − 3·9-s + 11-s + 4.47·13-s + 4.47·17-s + 2.47·19-s − 6.47·23-s + 25-s − 4.47·29-s + 6.47·31-s + 35-s − 4.47·37-s − 2·41-s − 4·43-s + 3·45-s + 2.47·47-s + 49-s + 8.47·53-s − 55-s − 1.52·59-s − 4.47·61-s + 3·63-s − 4.47·65-s − 12.9·67-s + 7.52·73-s − 77-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 9-s + 0.301·11-s + 1.24·13-s + 1.08·17-s + 0.567·19-s − 1.34·23-s + 0.200·25-s − 0.830·29-s + 1.16·31-s + 0.169·35-s − 0.735·37-s − 0.312·41-s − 0.609·43-s + 0.447·45-s + 0.360·47-s + 0.142·49-s + 1.16·53-s − 0.134·55-s − 0.198·59-s − 0.572·61-s + 0.377·63-s − 0.554·65-s − 1.58·67-s + 0.881·73-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.516300060\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.516300060\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 7.52T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−8.094344028898204548724967188905, −7.47841881979692677070518554860, −6.52785138149006083098260652526, −5.91378985305989834853973222072, −5.39551449022148277519209761652, −4.27624186926950911299816675905, −3.50863487871347615927752887624, −3.04588413031692263863172378566, −1.78317364699155316393025354613, −0.64694323777064887843649271325,
0.64694323777064887843649271325, 1.78317364699155316393025354613, 3.04588413031692263863172378566, 3.50863487871347615927752887624, 4.27624186926950911299816675905, 5.39551449022148277519209761652, 5.91378985305989834853973222072, 6.52785138149006083098260652526, 7.47841881979692677070518554860, 8.094344028898204548724967188905