| L(s) = 1 | + 5-s + 7-s − 3·9-s − 2·17-s − 6·19-s − 3·23-s − 4·25-s − 3·29-s − 2·31-s + 35-s − 6·37-s + 3·41-s + 7·43-s − 3·45-s − 6·49-s − 4·53-s + 3·59-s + 61-s − 3·63-s − 13·67-s − 10·71-s − 11·73-s − 8·79-s + 9·81-s − 4·83-s − 2·85-s + 12·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 9-s − 0.485·17-s − 1.37·19-s − 0.625·23-s − 4/5·25-s − 0.557·29-s − 0.359·31-s + 0.169·35-s − 0.986·37-s + 0.468·41-s + 1.06·43-s − 0.447·45-s − 6/7·49-s − 0.549·53-s + 0.390·59-s + 0.128·61-s − 0.377·63-s − 1.58·67-s − 1.18·71-s − 1.28·73-s − 0.900·79-s + 81-s − 0.439·83-s − 0.216·85-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.20277878787163, −12.69190565479956, −11.95187365607224, −11.80751670103576, −11.19093485444201, −10.83477123352989, −10.35599566231158, −9.989230719955165, −9.202002903041665, −9.001506556140346, −8.532886411251115, −8.074180576280877, −7.507909579391992, −7.141172774980814, −6.283379761515973, −6.063523208734356, −5.735391274748204, −5.034916926205293, −4.552041637208296, −4.050227455891648, −3.466461134263466, −2.857218967342401, −2.153142029348112, −1.965390444232949, −1.201216950918740, 0, 0,
1.201216950918740, 1.965390444232949, 2.153142029348112, 2.857218967342401, 3.466461134263466, 4.050227455891648, 4.552041637208296, 5.034916926205293, 5.735391274748204, 6.063523208734356, 6.283379761515973, 7.141172774980814, 7.507909579391992, 8.074180576280877, 8.532886411251115, 9.001506556140346, 9.202002903041665, 9.989230719955165, 10.35599566231158, 10.83477123352989, 11.19093485444201, 11.80751670103576, 11.95187365607224, 12.69190565479956, 13.20277878787163
