L(s) = 1 | + 3·3-s − 4·7-s + 6·9-s − 11-s + 3·13-s − 2·17-s + 4·19-s − 12·21-s + 6·23-s + 9·27-s − 29-s + 9·31-s − 3·33-s + 8·37-s + 9·39-s − 8·41-s + 5·43-s + 7·47-s + 9·49-s − 6·51-s + 5·53-s + 12·57-s − 10·59-s + 10·61-s − 24·63-s − 8·67-s + 18·69-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 1.51·7-s + 2·9-s − 0.301·11-s + 0.832·13-s − 0.485·17-s + 0.917·19-s − 2.61·21-s + 1.25·23-s + 1.73·27-s − 0.185·29-s + 1.61·31-s − 0.522·33-s + 1.31·37-s + 1.44·39-s − 1.24·41-s + 0.762·43-s + 1.02·47-s + 9/7·49-s − 0.840·51-s + 0.686·53-s + 1.58·57-s − 1.30·59-s + 1.28·61-s − 3.02·63-s − 0.977·67-s + 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.201218412\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.201218412\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 29 | \( 1 + T \) | |
good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−8.879395748073981170118839519763, −8.141025642923753279087882974777, −7.33550264153574239993784581896, −6.71069646532965173957055154462, −5.86667719704416489044012817015, −4.57978403736801959392711577274, −3.68228666659704414023251144515, −3.04081723023151694497231831115, −2.50541091856436250305362274671, −1.05600136943034290088228750938,
1.05600136943034290088228750938, 2.50541091856436250305362274671, 3.04081723023151694497231831115, 3.68228666659704414023251144515, 4.57978403736801959392711577274, 5.86667719704416489044012817015, 6.71069646532965173957055154462, 7.33550264153574239993784581896, 8.141025642923753279087882974777, 8.879395748073981170118839519763