| L(s) = 1 | + 3-s + 9-s − 3·13-s − 2·17-s − 19-s − 2·23-s + 27-s + 8·29-s − 8·31-s + 7·37-s − 3·39-s − 8·43-s + 10·47-s − 2·51-s − 14·53-s − 57-s − 10·59-s − 7·61-s − 5·67-s − 2·69-s − 12·71-s + 11·73-s − 7·79-s + 81-s + 14·83-s + 8·87-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.832·13-s − 0.485·17-s − 0.229·19-s − 0.417·23-s + 0.192·27-s + 1.48·29-s − 1.43·31-s + 1.15·37-s − 0.480·39-s − 1.21·43-s + 1.45·47-s − 0.280·51-s − 1.92·53-s − 0.132·57-s − 1.30·59-s − 0.896·61-s − 0.610·67-s − 0.240·69-s − 1.42·71-s + 1.28·73-s − 0.787·79-s + 1/9·81-s + 1.53·83-s + 0.857·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.707650427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707650427\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−12.94734277390936, −12.41221340733739, −12.15107460054037, −11.50924678868773, −11.04466270423789, −10.50319279868415, −10.11433456508837, −9.622508540625570, −9.049604753252625, −8.857705073227629, −8.158602940845143, −7.640486808753123, −7.446562591247828, −6.695892598977672, −6.285453854750855, −5.799014501379981, −5.004896743531005, −4.609358520555169, −4.233553214443902, −3.428108878745704, −3.032084699591002, −2.399467274611692, −1.909776628057457, −1.261225914359309, −0.3387621172619059,
0.3387621172619059, 1.261225914359309, 1.909776628057457, 2.399467274611692, 3.032084699591002, 3.428108878745704, 4.233553214443902, 4.609358520555169, 5.004896743531005, 5.799014501379981, 6.285453854750855, 6.695892598977672, 7.446562591247828, 7.640486808753123, 8.158602940845143, 8.857705073227629, 9.049604753252625, 9.622508540625570, 10.11433456508837, 10.50319279868415, 11.04466270423789, 11.50924678868773, 12.15107460054037, 12.41221340733739, 12.94734277390936
