| L(s) = 1 | + 2i·5-s + 8i·7-s + 4·11-s − 14i·13-s − 18·17-s + 12·19-s + 40i·23-s + 21·25-s + 14i·29-s − 32i·31-s − 16·35-s + 30i·37-s − 14·41-s + 28·43-s − 16i·47-s + ⋯ |
| L(s) = 1 | + 0.400i·5-s + 1.14i·7-s + 0.363·11-s − 1.07i·13-s − 1.05·17-s + 0.631·19-s + 1.73i·23-s + 0.839·25-s + 0.482i·29-s − 1.03i·31-s − 0.457·35-s + 0.810i·37-s − 0.341·41-s + 0.651·43-s − 0.340i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.377021252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.377021252\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2iT - 25T^{2} \) |
| 7 | \( 1 - 8iT - 49T^{2} \) |
| 11 | \( 1 - 4T + 121T^{2} \) |
| 13 | \( 1 + 14iT - 169T^{2} \) |
| 17 | \( 1 + 18T + 289T^{2} \) |
| 19 | \( 1 - 12T + 361T^{2} \) |
| 23 | \( 1 - 40iT - 529T^{2} \) |
| 29 | \( 1 - 14iT - 841T^{2} \) |
| 31 | \( 1 + 32iT - 961T^{2} \) |
| 37 | \( 1 - 30iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 14T + 1.68e3T^{2} \) |
| 43 | \( 1 - 28T + 1.84e3T^{2} \) |
| 47 | \( 1 + 16iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 66iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 52T + 3.48e3T^{2} \) |
| 61 | \( 1 - 82iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 56iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 66T + 5.32e3T^{2} \) |
| 79 | \( 1 + 16iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 140T + 6.88e3T^{2} \) |
| 89 | \( 1 + 30T + 7.92e3T^{2} \) |
| 97 | \( 1 + 14T + 9.40e3T^{2} \) |
| 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
−9.060714480967077384986164550781, −8.464354742660311588112614134130, −7.52813796594117754116480388757, −6.84627407533303238271333037674, −5.78782984936074719195243181819, −5.43801382566198138191347627896, −4.28028029658507749168266796459, −3.16624179055811715560990854326, −2.53481561775230157115243610200, −1.27980813534481900663269430524,
0.34989281455098309243453279081, 1.40685001618498820329014534203, 2.56994262751540823952920347494, 3.87450988093561287763792746996, 4.41014792053344680013675243790, 5.17486621881442857079067037867, 6.64417677792203423865510263904, 6.70479724491853084954998419344, 7.74418809318718047619012766589, 8.694626857682880956613881407611
