| L(s) = 1 | − 3-s − 7-s + 9-s + 4·13-s − 17-s − 4·19-s + 21-s − 27-s + 8·31-s + 10·37-s − 4·39-s + 6·41-s − 2·43-s + 6·47-s + 49-s + 51-s + 4·57-s − 12·59-s + 8·61-s − 63-s − 2·67-s − 12·71-s + 10·73-s + 2·79-s + 81-s − 6·83-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.10·13-s − 0.242·17-s − 0.917·19-s + 0.218·21-s − 0.192·27-s + 1.43·31-s + 1.64·37-s − 0.640·39-s + 0.937·41-s − 0.304·43-s + 0.875·47-s + 1/7·49-s + 0.140·51-s + 0.529·57-s − 1.56·59-s + 1.02·61-s − 0.125·63-s − 0.244·67-s − 1.42·71-s + 1.17·73-s + 0.225·79-s + 1/9·81-s − 0.658·83-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.776935479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.776935479\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.09761585391549, −14.32251306209209, −13.85579910433871, −13.17800984735668, −12.92236042468305, −12.33932884294275, −11.64186634179522, −11.29850975383825, −10.65771537079720, −10.30676305485210, −9.610079827270637, −9.057968586527463, −8.476725952332330, −7.924700914692567, −7.260133827065029, −6.541653216120201, −6.098197571796220, −5.813041520315024, −4.781927808897187, −4.351093086570836, −3.740458602052321, −2.909740409308538, −2.231691342617984, −1.255430581298009, −0.5653520501879424,
0.5653520501879424, 1.255430581298009, 2.231691342617984, 2.909740409308538, 3.740458602052321, 4.351093086570836, 4.781927808897187, 5.813041520315024, 6.098197571796220, 6.541653216120201, 7.260133827065029, 7.924700914692567, 8.476725952332330, 9.057968586527463, 9.610079827270637, 10.30676305485210, 10.65771537079720, 11.29850975383825, 11.64186634179522, 12.33932884294275, 12.92236042468305, 13.17800984735668, 13.85579910433871, 14.32251306209209, 15.09761585391549
