| L(s) = 1 | − 3-s + 7-s + 9-s + 11-s − 13-s − 4·17-s + 3·19-s − 21-s − 6·23-s − 27-s + 7·29-s + 4·31-s − 33-s − 37-s + 39-s − 4·41-s + 2·43-s − 7·47-s + 49-s + 4·51-s − 10·53-s − 3·57-s − 9·59-s − 2·61-s + 63-s + 9·67-s + 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 0.970·17-s + 0.688·19-s − 0.218·21-s − 1.25·23-s − 0.192·27-s + 1.29·29-s + 0.718·31-s − 0.174·33-s − 0.164·37-s + 0.160·39-s − 0.624·41-s + 0.304·43-s − 1.02·47-s + 1/7·49-s + 0.560·51-s − 1.37·53-s − 0.397·57-s − 1.17·59-s − 0.256·61-s + 0.125·63-s + 1.09·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.494894010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494894010\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 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.ae |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 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.k |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 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.e |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 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.ao |
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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
−15.72309129740363, −15.01821063289652, −14.16715568618929, −14.03056244099139, −13.32924006956493, −12.68422757772854, −11.98565252765305, −11.84774135246426, −11.10790254089083, −10.62972396082792, −9.976372390520079, −9.518634295025527, −8.828307088453646, −8.080707433645977, −7.764267454602228, −6.801694089486024, −6.463369504894282, −5.861877498056565, −4.914544630168278, −4.722235216992882, −3.897070458471122, −3.106829513210466, −2.212375926696936, −1.495768851534657, −0.5188218761600296,
0.5188218761600296, 1.495768851534657, 2.212375926696936, 3.106829513210466, 3.897070458471122, 4.722235216992882, 4.914544630168278, 5.861877498056565, 6.463369504894282, 6.801694089486024, 7.764267454602228, 8.080707433645977, 8.828307088453646, 9.518634295025527, 9.976372390520079, 10.62972396082792, 11.10790254089083, 11.84774135246426, 11.98565252765305, 12.68422757772854, 13.32924006956493, 14.03056244099139, 14.16715568618929, 15.01821063289652, 15.72309129740363
