| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s + 9-s − 2·12-s − 4·16-s − 2·17-s − 2·18-s + 5·19-s − 6·23-s − 5·25-s − 27-s − 4·29-s + 8·32-s + 4·34-s + 2·36-s + 3·37-s − 10·38-s + 11·43-s + 12·46-s + 4·47-s + 4·48-s − 7·49-s + 10·50-s + 2·51-s − 12·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s − 0.577·12-s − 16-s − 0.485·17-s − 0.471·18-s + 1.14·19-s − 1.25·23-s − 25-s − 0.192·27-s − 0.742·29-s + 1.41·32-s + 0.685·34-s + 1/3·36-s + 0.493·37-s − 1.62·38-s + 1.67·43-s + 1.76·46-s + 0.583·47-s + 0.577·48-s − 49-s + 1.41·50-s + 0.280·51-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3254664728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3254664728\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
| 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
−14.22399244996987, −13.77800412264214, −13.29825243392947, −12.66283165381416, −11.95211765134026, −11.72284559689372, −10.91077302578464, −10.86334493383090, −10.06385359199035, −9.583452429941196, −9.350169950826082, −8.706849484569746, −7.986300524503986, −7.526337369393024, −7.382456664401531, −6.329848753960665, −6.118712962088336, −5.390693397301541, −4.622077599435903, −4.152008782609958, −3.350051730720296, −2.447151444591747, −1.783311138293646, −1.190105452145558, −0.2769639517967503,
0.2769639517967503, 1.190105452145558, 1.783311138293646, 2.447151444591747, 3.350051730720296, 4.152008782609958, 4.622077599435903, 5.390693397301541, 6.118712962088336, 6.329848753960665, 7.382456664401531, 7.526337369393024, 7.986300524503986, 8.706849484569746, 9.350169950826082, 9.583452429941196, 10.06385359199035, 10.86334493383090, 10.91077302578464, 11.72284559689372, 11.95211765134026, 12.66283165381416, 13.29825243392947, 13.77800412264214, 14.22399244996987
