L(s) = 1 | − 3.99i·5-s − 7.74i·7-s + 7.68·11-s − 1.42i·13-s + 22.3·17-s + 21.9·19-s + 18.1i·23-s + 9.02·25-s − 15.2i·29-s + 10.2i·31-s − 30.9·35-s + 59.1i·37-s + 41.3·41-s + 11.0·43-s − 63.6i·47-s + ⋯ |
L(s) = 1 | − 0.799i·5-s − 1.10i·7-s + 0.699·11-s − 0.109i·13-s + 1.31·17-s + 1.15·19-s + 0.789i·23-s + 0.360·25-s − 0.527i·29-s + 0.329i·31-s − 0.884·35-s + 1.59i·37-s + 1.00·41-s + 0.255·43-s − 1.35i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.346735267\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.346735267\) |
\(L(2)\) |
|
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 | \( 1 + 3.99iT - 25T^{2} \) |
| 7 | \( 1 + 7.74iT - 49T^{2} \) |
| 11 | \( 1 - 7.68T + 121T^{2} \) |
| 13 | \( 1 + 1.42iT - 169T^{2} \) |
| 17 | \( 1 - 22.3T + 289T^{2} \) |
| 19 | \( 1 - 21.9T + 361T^{2} \) |
| 23 | \( 1 - 18.1iT - 529T^{2} \) |
| 29 | \( 1 + 15.2iT - 841T^{2} \) |
| 31 | \( 1 - 10.2iT - 961T^{2} \) |
| 37 | \( 1 - 59.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 41.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 11.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 63.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 19.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 16.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 27.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 60.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 134. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 90.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 12.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 164.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 35.6T + 7.92e3T^{2} \) |
| 97 | \( 1 + 147.T + 9.40e3T^{2} \) |
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
−9.054806879384700164008245871615, −8.070831208351196004960339973372, −7.49272057130247494462363143368, −6.66492858798364885445108282469, −5.59671425125024474371904657590, −4.86722035873012751635601885986, −3.91409066123994360684078679687, −3.16278081831135029553097312990, −1.42209983941032298892833949482, −0.78725293874372804481801919003,
1.12791845915317175462117261669, 2.49489127102071754915611727844, 3.17418212278838751162737892827, 4.24303211075874005763891989869, 5.49500795510466149335939057948, 5.94816139111995562525834523073, 6.97728470484019010243246588284, 7.60579625629709978858520087561, 8.618189044266336190706782717482, 9.313339817285487216669218349948