| L(s) = 1 | + 5-s − 2·7-s − 3·11-s − 13-s + 17-s − 3·19-s − 3·23-s − 4·25-s + 2·29-s − 4·31-s − 2·35-s − 6·37-s + 3·41-s − 5·43-s − 3·49-s + 8·53-s − 3·55-s − 2·59-s − 10·61-s − 65-s + 4·67-s + 8·71-s + 4·73-s + 6·77-s − 10·79-s − 16·83-s + 85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.904·11-s − 0.277·13-s + 0.242·17-s − 0.688·19-s − 0.625·23-s − 4/5·25-s + 0.371·29-s − 0.718·31-s − 0.338·35-s − 0.986·37-s + 0.468·41-s − 0.762·43-s − 3/7·49-s + 1.09·53-s − 0.404·55-s − 0.260·59-s − 1.28·61-s − 0.124·65-s + 0.488·67-s + 0.949·71-s + 0.468·73-s + 0.683·77-s − 1.12·79-s − 1.75·83-s + 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−9.459623231881873822356567961753, −8.513820787940616730761937857356, −7.68974868342096212093047927681, −6.77543845197466226845406846618, −5.93758174354952614509205800358, −5.18833949456990331172943189244, −4.02121845958285722319390753632, −2.95216897524886256164197886588, −1.92607727089333104341553701985, 0,
1.92607727089333104341553701985, 2.95216897524886256164197886588, 4.02121845958285722319390753632, 5.18833949456990331172943189244, 5.93758174354952614509205800358, 6.77543845197466226845406846618, 7.68974868342096212093047927681, 8.513820787940616730761937857356, 9.459623231881873822356567961753
