| L(s) = 1 | + 3-s + 9-s + 2·11-s + 13-s − 8·17-s − 7·19-s + 6·23-s + 27-s − 4·29-s + 8·31-s + 2·33-s + 7·37-s + 39-s + 2·41-s + 4·43-s − 12·47-s − 8·51-s − 4·53-s − 7·57-s − 12·59-s − 3·61-s + 9·67-s + 6·69-s − 12·71-s − 9·73-s − 17·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 1.94·17-s − 1.60·19-s + 1.25·23-s + 0.192·27-s − 0.742·29-s + 1.43·31-s + 0.348·33-s + 1.15·37-s + 0.160·39-s + 0.312·41-s + 0.609·43-s − 1.75·47-s − 1.12·51-s − 0.549·53-s − 0.927·57-s − 1.56·59-s − 0.384·61-s + 1.09·67-s + 0.722·69-s − 1.42·71-s − 1.05·73-s − 1.91·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.247069158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.247069158\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.02146793090954, −12.73068124690952, −11.91911270357681, −11.34293219350302, −11.18873405151846, −10.59216830441690, −10.16901167807182, −9.437735520277163, −9.147170037866413, −8.653028794593129, −8.434619783800011, −7.715548050150722, −7.239403031974836, −6.668920840873537, −6.175991243109471, −6.078393728647449, −4.879118140470863, −4.533966483204897, −4.295960743042997, −3.539102661169860, −2.952313759498427, −2.420723768565822, −1.869696340223837, −1.272940994941258, −0.3910668553485669,
0.3910668553485669, 1.272940994941258, 1.869696340223837, 2.420723768565822, 2.952313759498427, 3.539102661169860, 4.295960743042997, 4.533966483204897, 4.879118140470863, 6.078393728647449, 6.175991243109471, 6.668920840873537, 7.239403031974836, 7.715548050150722, 8.434619783800011, 8.653028794593129, 9.147170037866413, 9.437735520277163, 10.16901167807182, 10.59216830441690, 11.18873405151846, 11.34293219350302, 11.91911270357681, 12.73068124690952, 13.02146793090954
