| L(s) = 1 | − 3-s − 2·5-s + 7-s − 2·9-s + 2·15-s − 3·17-s − 7·19-s − 21-s − 23-s − 25-s + 5·27-s − 3·29-s + 8·31-s − 2·35-s + 37-s + 11·41-s + 11·43-s + 4·45-s + 12·47-s − 6·49-s + 3·51-s − 6·53-s + 7·57-s − 9·59-s + 9·61-s − 2·63-s − 3·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s − 2/3·9-s + 0.516·15-s − 0.727·17-s − 1.60·19-s − 0.218·21-s − 0.208·23-s − 1/5·25-s + 0.962·27-s − 0.557·29-s + 1.43·31-s − 0.338·35-s + 0.164·37-s + 1.71·41-s + 1.67·43-s + 0.596·45-s + 1.75·47-s − 6/7·49-s + 0.420·51-s − 0.824·53-s + 0.927·57-s − 1.17·59-s + 1.15·61-s − 0.251·63-s − 0.366·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.271362623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.271362623\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−12.60243431038631, −12.09164073652099, −11.62214877165660, −11.18934799402411, −10.99262600429282, −10.53163612103835, −9.992713809194947, −9.315446305338124, −8.799086874249761, −8.514055166982630, −8.031858428359690, −7.414531197965639, −7.248436276741408, −6.283151358726411, −6.078279969059948, −5.797031020313050, −4.832932052780567, −4.509037632948647, −4.209464428520225, −3.580787102870751, −2.834487499821873, −2.372814176224929, −1.811268437334258, −0.7767441143173918, −0.4218994175118649,
0.4218994175118649, 0.7767441143173918, 1.811268437334258, 2.372814176224929, 2.834487499821873, 3.580787102870751, 4.209464428520225, 4.509037632948647, 4.832932052780567, 5.797031020313050, 6.078279969059948, 6.283151358726411, 7.248436276741408, 7.414531197965639, 8.031858428359690, 8.514055166982630, 8.799086874249761, 9.315446305338124, 9.992713809194947, 10.53163612103835, 10.99262600429282, 11.18934799402411, 11.62214877165660, 12.09164073652099, 12.60243431038631
