| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 11-s + 13-s + 15-s + 4·17-s − 3·19-s − 21-s − 6·23-s − 4·25-s − 27-s + 7·29-s − 4·31-s + 33-s − 35-s + 37-s − 39-s − 4·41-s + 2·43-s − 45-s − 7·47-s + 49-s − 4·51-s + 10·53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.258·15-s + 0.970·17-s − 0.688·19-s − 0.218·21-s − 1.25·23-s − 4/5·25-s − 0.192·27-s + 1.29·29-s − 0.718·31-s + 0.174·33-s − 0.169·35-s + 0.164·37-s − 0.160·39-s − 0.624·41-s + 0.304·43-s − 0.149·45-s − 1.02·47-s + 1/7·49-s − 0.560·51-s + 1.37·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−8.142769749401240638842099218410, −7.48081830346601360916973270588, −6.64154799609352895225991337192, −5.84727569254735575475982369360, −5.22535186346629942841947571426, −4.27400667077812408101837638391, −3.66630303675510058944233818355, −2.43871280217448199990766691349, −1.32780456660572558565910750326, 0,
1.32780456660572558565910750326, 2.43871280217448199990766691349, 3.66630303675510058944233818355, 4.27400667077812408101837638391, 5.22535186346629942841947571426, 5.84727569254735575475982369360, 6.64154799609352895225991337192, 7.48081830346601360916973270588, 8.142769749401240638842099218410
