| L(s) = 1 | − 2·7-s − 11-s − 13-s + 8·17-s − 5·19-s − 9·23-s + 5·29-s − 7·31-s − 2·37-s − 2·41-s − 9·43-s + 12·47-s − 3·49-s − 4·53-s + 10·59-s − 2·61-s + 2·67-s − 13·71-s − 14·73-s + 2·77-s − 83-s + 15·89-s + 2·91-s + 7·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.301·11-s − 0.277·13-s + 1.94·17-s − 1.14·19-s − 1.87·23-s + 0.928·29-s − 1.25·31-s − 0.328·37-s − 0.312·41-s − 1.37·43-s + 1.75·47-s − 3/7·49-s − 0.549·53-s + 1.30·59-s − 0.256·61-s + 0.244·67-s − 1.54·71-s − 1.63·73-s + 0.227·77-s − 0.109·83-s + 1.58·89-s + 0.209·91-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6012940571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6012940571\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
| 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
−13.27675419265559, −12.70978242517467, −12.38791711880603, −11.89705003147561, −11.61980748136328, −10.66282135393005, −10.35706711000788, −10.01373659623380, −9.631132766776097, −8.897336914083140, −8.480635089245116, −7.930761878791955, −7.483928553102808, −7.000782817526599, −6.241861179981577, −6.023455027794772, −5.411251629007551, −4.900272613980753, −4.125497976870077, −3.694543942381957, −3.174210059042861, −2.532240294937544, −1.898749033956015, −1.226028836268645, −0.2282814345527808,
0.2282814345527808, 1.226028836268645, 1.898749033956015, 2.532240294937544, 3.174210059042861, 3.694543942381957, 4.125497976870077, 4.900272613980753, 5.411251629007551, 6.023455027794772, 6.241861179981577, 7.000782817526599, 7.483928553102808, 7.930761878791955, 8.480635089245116, 8.897336914083140, 9.631132766776097, 10.01373659623380, 10.35706711000788, 10.66282135393005, 11.61980748136328, 11.89705003147561, 12.38791711880603, 12.70978242517467, 13.27675419265559
