| L(s) = 1 | + 5-s − 2·7-s − 13-s − 3·17-s − 5·19-s + 6·23-s + 25-s − 6·29-s + 2·31-s − 2·35-s + 5·37-s + 9·41-s + 43-s + 3·47-s − 3·49-s + 12·59-s − 8·61-s − 65-s + 5·67-s − 12·71-s − 2·73-s + 10·79-s − 6·83-s − 3·85-s + 2·91-s − 5·95-s − 97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.277·13-s − 0.727·17-s − 1.14·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s + 0.359·31-s − 0.338·35-s + 0.821·37-s + 1.40·41-s + 0.152·43-s + 0.437·47-s − 3/7·49-s + 1.56·59-s − 1.02·61-s − 0.124·65-s + 0.610·67-s − 1.42·71-s − 0.234·73-s + 1.12·79-s − 0.658·83-s − 0.325·85-s + 0.209·91-s − 0.512·95-s − 0.101·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−13.02350392499831, −12.64263288373774, −12.18036105833692, −11.49455037334300, −11.05583151014361, −10.71933153957618, −10.26191691762516, −9.514796849991498, −9.428218236257761, −8.929203602342205, −8.382351900982870, −7.880065072404980, −7.164539651966142, −6.924030780441862, −6.351380674161802, −5.926117716304671, −5.475048353925091, −4.796973756244179, −4.307636527509694, −3.872280890253833, −3.119087701449196, −2.612762488249250, −2.217936086292099, −1.468347763854084, −0.7147523413817716, 0,
0.7147523413817716, 1.468347763854084, 2.217936086292099, 2.612762488249250, 3.119087701449196, 3.872280890253833, 4.307636527509694, 4.796973756244179, 5.475048353925091, 5.926117716304671, 6.351380674161802, 6.924030780441862, 7.164539651966142, 7.880065072404980, 8.382351900982870, 8.929203602342205, 9.428218236257761, 9.514796849991498, 10.26191691762516, 10.71933153957618, 11.05583151014361, 11.49455037334300, 12.18036105833692, 12.64263288373774, 13.02350392499831
