| L(s) = 1 | + 5-s − 2·11-s − 13-s + 17-s − 8·19-s − 6·23-s + 25-s − 4·31-s + 2·37-s − 4·41-s − 4·43-s − 7·49-s + 6·53-s − 2·55-s − 4·59-s + 8·61-s − 65-s + 8·67-s + 4·71-s + 10·79-s − 8·83-s + 85-s − 6·89-s − 8·95-s + 8·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.603·11-s − 0.277·13-s + 0.242·17-s − 1.83·19-s − 1.25·23-s + 1/5·25-s − 0.718·31-s + 0.328·37-s − 0.624·41-s − 0.609·43-s − 49-s + 0.824·53-s − 0.269·55-s − 0.520·59-s + 1.02·61-s − 0.124·65-s + 0.977·67-s + 0.474·71-s + 1.12·79-s − 0.878·83-s + 0.108·85-s − 0.635·89-s − 0.820·95-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9578361791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9578361791\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.13984522565794, −13.32150655561149, −13.13249866324483, −12.46479242713400, −12.23240575149508, −11.36442956712504, −11.07009073094496, −10.35659225089561, −10.05843076534120, −9.628084536830404, −8.841231378214705, −8.463889336577581, −7.935744882981305, −7.428427267976677, −6.601308832915235, −6.393788208852618, −5.659855275567428, −5.185467192925619, −4.580773773964677, −3.931834520413895, −3.403592170749322, −2.427099977918653, −2.178694021972665, −1.422462789976538, −0.3037766695015894,
0.3037766695015894, 1.422462789976538, 2.178694021972665, 2.427099977918653, 3.403592170749322, 3.931834520413895, 4.580773773964677, 5.185467192925619, 5.659855275567428, 6.393788208852618, 6.601308832915235, 7.428427267976677, 7.935744882981305, 8.463889336577581, 8.841231378214705, 9.628084536830404, 10.05843076534120, 10.35659225089561, 11.07009073094496, 11.36442956712504, 12.23240575149508, 12.46479242713400, 13.13249866324483, 13.32150655561149, 14.13984522565794
