| L(s) = 1 | − 3-s − 3·5-s − 7-s + 9-s + 11-s + 13-s + 3·15-s + 4·17-s + 19-s + 21-s + 4·23-s + 4·25-s − 27-s + 9·29-s − 2·31-s − 33-s + 3·35-s − 2·37-s − 39-s − 8·41-s − 3·45-s + 11·47-s − 6·49-s − 4·51-s + 4·53-s − 3·55-s − 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.774·15-s + 0.970·17-s + 0.229·19-s + 0.218·21-s + 0.834·23-s + 4/5·25-s − 0.192·27-s + 1.67·29-s − 0.359·31-s − 0.174·33-s + 0.507·35-s − 0.328·37-s − 0.160·39-s − 1.24·41-s − 0.447·45-s + 1.60·47-s − 6/7·49-s − 0.560·51-s + 0.549·53-s − 0.404·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−14.18526341368377, −13.59576310083533, −13.02351522576483, −12.46530997343554, −12.12440935611647, −11.67061943481260, −11.33806178869057, −10.72750858549572, −10.11116900512307, −9.922080768073167, −8.868435532078115, −8.736841196613709, −7.998572154945368, −7.510985015446446, −7.031718357575827, −6.550006081156882, −5.985415357409726, −5.182191526740928, −4.928455092458524, −4.056961808184549, −3.713399022944370, −3.154596220566869, −2.462269890765809, −1.303595010611750, −0.8416338592691745, 0,
0.8416338592691745, 1.303595010611750, 2.462269890765809, 3.154596220566869, 3.713399022944370, 4.056961808184549, 4.928455092458524, 5.182191526740928, 5.985415357409726, 6.550006081156882, 7.031718357575827, 7.510985015446446, 7.998572154945368, 8.736841196613709, 8.868435532078115, 9.922080768073167, 10.11116900512307, 10.72750858549572, 11.33806178869057, 11.67061943481260, 12.12440935611647, 12.46530997343554, 13.02351522576483, 13.59576310083533, 14.18526341368377
