| L(s) = 1 | + 2·5-s + 2·7-s + 4·11-s − 4·13-s − 7·17-s − 3·19-s − 25-s − 4·29-s − 6·31-s + 4·35-s + 2·37-s − 6·41-s − 5·43-s − 10·47-s − 3·49-s + 8·55-s + 5·59-s − 4·61-s − 8·65-s − 5·67-s + 14·71-s + 15·73-s + 8·77-s − 12·79-s − 15·83-s − 14·85-s − 10·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 1.20·11-s − 1.10·13-s − 1.69·17-s − 0.688·19-s − 1/5·25-s − 0.742·29-s − 1.07·31-s + 0.676·35-s + 0.328·37-s − 0.937·41-s − 0.762·43-s − 1.45·47-s − 3/7·49-s + 1.07·55-s + 0.650·59-s − 0.512·61-s − 0.992·65-s − 0.610·67-s + 1.66·71-s + 1.75·73-s + 0.911·77-s − 1.35·79-s − 1.64·83-s − 1.51·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.093108438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.093108438\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.63961722066279, −12.33098033401997, −11.56532705446983, −11.26596575066151, −11.08408840732021, −10.29991082449647, −9.830827695701436, −9.486473914943427, −9.084942374969898, −8.504150659919726, −8.220717399945583, −7.471800379498011, −6.978283546333245, −6.532487835678221, −6.280341980603963, −5.382329092488441, −5.231991253224026, −4.490363014688228, −4.191436734187408, −3.564147538907630, −2.797818416503699, −2.112578815015686, −1.827294253451408, −1.419370885710074, −0.2507158676354670,
0.2507158676354670, 1.419370885710074, 1.827294253451408, 2.112578815015686, 2.797818416503699, 3.564147538907630, 4.191436734187408, 4.490363014688228, 5.231991253224026, 5.382329092488441, 6.280341980603963, 6.532487835678221, 6.978283546333245, 7.471800379498011, 8.220717399945583, 8.504150659919726, 9.084942374969898, 9.486473914943427, 9.830827695701436, 10.29991082449647, 11.08408840732021, 11.26596575066151, 11.56532705446983, 12.33098033401997, 12.63961722066279
