| L(s) = 1 | − 2·3-s − 7-s + 9-s + 4·11-s − 2·17-s + 6·19-s + 2·21-s − 8·23-s − 5·25-s + 4·27-s − 2·29-s − 4·31-s − 8·33-s + 10·37-s + 10·41-s + 4·43-s + 4·47-s + 49-s + 4·51-s + 2·53-s − 12·57-s − 10·59-s + 8·61-s − 63-s + 8·67-s + 16·69-s + 6·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.485·17-s + 1.37·19-s + 0.436·21-s − 1.66·23-s − 25-s + 0.769·27-s − 0.371·29-s − 0.718·31-s − 1.39·33-s + 1.64·37-s + 1.56·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.560·51-s + 0.274·53-s − 1.58·57-s − 1.30·59-s + 1.02·61-s − 0.125·63-s + 0.977·67-s + 1.92·69-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.347332558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.347332558\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.98830427984110, −13.70770612408389, −12.85852723780420, −12.43303048858608, −12.01545056852027, −11.51660982995108, −11.19545443865495, −10.75334307628670, −9.909850764115871, −9.570710233395468, −9.243172769143080, −8.471427559138929, −7.729548661958670, −7.415358806465589, −6.592062348676450, −6.262366107348887, −5.697536507937985, −5.456858875104289, −4.471675793045535, −4.070331716915960, −3.555741097609860, −2.635595270584109, −1.957582212657458, −1.063760353841250, −0.4813387031434684,
0.4813387031434684, 1.063760353841250, 1.957582212657458, 2.635595270584109, 3.555741097609860, 4.070331716915960, 4.471675793045535, 5.456858875104289, 5.697536507937985, 6.262366107348887, 6.592062348676450, 7.415358806465589, 7.729548661958670, 8.471427559138929, 9.243172769143080, 9.570710233395468, 9.909850764115871, 10.75334307628670, 11.19545443865495, 11.51660982995108, 12.01545056852027, 12.43303048858608, 12.85852723780420, 13.70770612408389, 13.98830427984110
