| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 6·11-s + 2·13-s − 15-s − 4·17-s − 2·19-s − 2·21-s + 25-s − 27-s − 4·29-s + 8·31-s + 6·33-s + 2·35-s + 2·37-s − 2·39-s + 41-s + 4·43-s + 45-s + 6·47-s − 3·49-s + 4·51-s + 6·53-s − 6·55-s + 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s − 0.258·15-s − 0.970·17-s − 0.458·19-s − 0.436·21-s + 1/5·25-s − 0.192·27-s − 0.742·29-s + 1.43·31-s + 1.04·33-s + 0.338·35-s + 0.328·37-s − 0.320·39-s + 0.156·41-s + 0.609·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.560·51-s + 0.824·53-s − 0.809·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 41 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
| 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
−15.15973367341379, −14.66575481632245, −13.75494680030843, −13.48995189998197, −13.13450318402364, −12.41348472513463, −12.00599740973342, −11.15911538680858, −10.84191768545109, −10.58864555289312, −9.885434194925369, −9.265145045637525, −8.611696966223553, −8.022631828868835, −7.685817364560445, −6.865088402767752, −6.338938661559352, −5.615404821241354, −5.341788105917631, −4.512415543857609, −4.259629260005846, −3.094202012816987, −2.435907781855216, −1.877287384738927, −0.9343467229342689, 0,
0.9343467229342689, 1.877287384738927, 2.435907781855216, 3.094202012816987, 4.259629260005846, 4.512415543857609, 5.341788105917631, 5.615404821241354, 6.338938661559352, 6.865088402767752, 7.685817364560445, 8.022631828868835, 8.611696966223553, 9.265145045637525, 9.885434194925369, 10.58864555289312, 10.84191768545109, 11.15911538680858, 12.00599740973342, 12.41348472513463, 13.13450318402364, 13.48995189998197, 13.75494680030843, 14.66575481632245, 15.15973367341379
