| L(s) = 1 | − 2·5-s + 4·7-s − 4·11-s + 4·17-s + 4·19-s − 4·23-s − 25-s + 2·29-s − 4·31-s − 8·35-s − 12·37-s − 12·41-s + 8·43-s + 9·49-s − 14·53-s + 8·55-s + 2·59-s − 2·61-s + 4·67-s + 8·71-s + 6·73-s − 16·77-s + 14·79-s + 6·83-s − 8·85-s + 6·89-s − 8·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 1.20·11-s + 0.970·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s + 0.371·29-s − 0.718·31-s − 1.35·35-s − 1.97·37-s − 1.87·41-s + 1.21·43-s + 9/7·49-s − 1.92·53-s + 1.07·55-s + 0.260·59-s − 0.256·61-s + 0.488·67-s + 0.949·71-s + 0.702·73-s − 1.82·77-s + 1.57·79-s + 0.658·83-s − 0.867·85-s + 0.635·89-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.650594005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.650594005\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.46230309788117, −14.91031836020298, −14.34366933707070, −13.86843503411682, −13.45864967475271, −12.40777481736532, −12.17174278881113, −11.71981727114738, −10.95332881993495, −10.74537990707519, −10.00621831562651, −9.404775046375501, −8.416421065412673, −8.194761855800519, −7.619585577583873, −7.369940228807036, −6.400403539819234, −5.375901760601492, −5.216481556285921, −4.590989355514131, −3.656975308181494, −3.285308094139035, −2.175397622012844, −1.580546324929874, −0.5153285748921119,
0.5153285748921119, 1.580546324929874, 2.175397622012844, 3.285308094139035, 3.656975308181494, 4.590989355514131, 5.216481556285921, 5.375901760601492, 6.400403539819234, 7.369940228807036, 7.619585577583873, 8.194761855800519, 8.416421065412673, 9.404775046375501, 10.00621831562651, 10.74537990707519, 10.95332881993495, 11.71981727114738, 12.17174278881113, 12.40777481736532, 13.45864967475271, 13.86843503411682, 14.34366933707070, 14.91031836020298, 15.46230309788117
