| L(s) = 1 | − 2·3-s − 2·5-s + 9-s + 4·11-s + 4·15-s + 6·17-s − 8·19-s − 25-s + 4·27-s − 2·29-s − 6·31-s − 8·33-s + 2·37-s − 8·43-s − 2·45-s − 2·47-s − 12·51-s + 6·53-s − 8·55-s + 16·57-s − 6·59-s + 10·61-s − 12·67-s + 16·71-s − 4·73-s + 2·75-s + 4·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s + 1.03·15-s + 1.45·17-s − 1.83·19-s − 1/5·25-s + 0.769·27-s − 0.371·29-s − 1.07·31-s − 1.39·33-s + 0.328·37-s − 1.21·43-s − 0.298·45-s − 0.291·47-s − 1.68·51-s + 0.824·53-s − 1.07·55-s + 2.11·57-s − 0.781·59-s + 1.28·61-s − 1.46·67-s + 1.89·71-s − 0.468·73-s + 0.230·75-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−13.07478845298268, −12.60173025796451, −12.10278751720725, −11.96904421333817, −11.35290084890915, −11.14886680853897, −10.50145960995586, −10.20259106330894, −9.497675447443666, −9.040400282718449, −8.461944373391422, −8.014696064773816, −7.548577615564859, −6.818045107106211, −6.591591585587927, −6.032434204987664, −5.521379270645257, −5.072312947835617, −4.392098561603824, −3.895100997723970, −3.610880130688449, −2.835065207237583, −1.944583038519747, −1.368724841965489, −0.6104198646755134, 0,
0.6104198646755134, 1.368724841965489, 1.944583038519747, 2.835065207237583, 3.610880130688449, 3.895100997723970, 4.392098561603824, 5.072312947835617, 5.521379270645257, 6.032434204987664, 6.591591585587927, 6.818045107106211, 7.548577615564859, 8.014696064773816, 8.461944373391422, 9.040400282718449, 9.497675447443666, 10.20259106330894, 10.50145960995586, 11.14886680853897, 11.35290084890915, 11.96904421333817, 12.10278751720725, 12.60173025796451, 13.07478845298268
