| L(s) = 1 | + i·5-s + 3.46i·11-s + 3.46i·13-s − 3.46·17-s + 4i·19-s − 25-s − 6i·29-s + 3.46·31-s − 3.46i·37-s − 6.92·41-s − 4i·43-s − 12·47-s − 7·49-s − 6i·53-s − 3.46·55-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 1.04i·11-s + 0.960i·13-s − 0.840·17-s + 0.917i·19-s − 0.200·25-s − 1.11i·29-s + 0.622·31-s − 0.569i·37-s − 1.08·41-s − 0.609i·43-s − 1.75·47-s − 49-s − 0.824i·53-s − 0.467·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7604044733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7604044733\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 + 3.46iT - 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 - 13.8iT - 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
| 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
−9.196061920243903886041078215175, −8.305436123891216798998893805687, −7.59698814453012990822646995658, −6.71655092236552881062633223705, −6.35051995647400462568826219025, −5.17246276347372053012988691613, −4.37861499620340767171492589987, −3.64831149060537691790954595544, −2.40646412631811534674645934426, −1.68591569063055145264997413258,
0.23417480768093765063143128891, 1.43634286356530895594028325308, 2.81303029260769667657205135983, 3.46514764749875631524545818920, 4.70847182709817942351490025705, 5.18683837436781675448449307285, 6.20951248550338930300086343201, 6.78527210121725119072200207851, 7.88402551763727822862772173057, 8.433925461723189430486701660380
