| L(s) = 1 | − 3-s − 3·5-s − 7-s + 9-s − 11-s + 3·15-s + 7·17-s − 6·19-s + 21-s + 4·23-s + 4·25-s − 27-s − 10·29-s + 33-s + 3·35-s + 6·37-s + 41-s + 10·43-s − 3·45-s + 3·47-s − 6·49-s − 7·51-s − 11·53-s + 3·55-s + 6·57-s − 8·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.774·15-s + 1.69·17-s − 1.37·19-s + 0.218·21-s + 0.834·23-s + 4/5·25-s − 0.192·27-s − 1.85·29-s + 0.174·33-s + 0.507·35-s + 0.986·37-s + 0.156·41-s + 1.52·43-s − 0.447·45-s + 0.437·47-s − 6/7·49-s − 0.980·51-s − 1.51·53-s + 0.404·55-s + 0.794·57-s − 1.04·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 356928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4858855253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4858855253\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−12.46282494761943, −12.31654981034299, −11.41591771851334, −11.22464473952646, −10.90345323958490, −10.44997656068342, −9.703570009236992, −9.465032611850517, −8.957749278133586, −8.136226745171856, −7.886380654699420, −7.631929380756413, −7.002581668140537, −6.522659264448729, −6.037213944711560, −5.436803128600882, −5.109804357081736, −4.392966895084564, −3.934679040967776, −3.633183508486786, −2.966324176596660, −2.437340139597529, −1.556937304803013, −0.9397861186615023, −0.2284767417233003,
0.2284767417233003, 0.9397861186615023, 1.556937304803013, 2.437340139597529, 2.966324176596660, 3.633183508486786, 3.934679040967776, 4.392966895084564, 5.109804357081736, 5.436803128600882, 6.037213944711560, 6.522659264448729, 7.002581668140537, 7.631929380756413, 7.886380654699420, 8.136226745171856, 8.957749278133586, 9.465032611850517, 9.703570009236992, 10.44997656068342, 10.90345323958490, 11.22464473952646, 11.41591771851334, 12.31654981034299, 12.46282494761943
