| L(s) = 1 | − 5-s − 3·13-s + 7·17-s + 4·19-s + 8·23-s − 4·25-s + 29-s − 8·31-s − 9·37-s + 10·41-s + 4·43-s − 8·47-s + 10·53-s + 12·59-s + 5·61-s + 3·65-s − 12·67-s − 7·73-s + 8·79-s − 7·85-s + 7·89-s − 4·95-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.832·13-s + 1.69·17-s + 0.917·19-s + 1.66·23-s − 4/5·25-s + 0.185·29-s − 1.43·31-s − 1.47·37-s + 1.56·41-s + 0.609·43-s − 1.16·47-s + 1.37·53-s + 1.56·59-s + 0.640·61-s + 0.372·65-s − 1.46·67-s − 0.819·73-s + 0.900·79-s − 0.759·85-s + 0.741·89-s − 0.410·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.232532478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.232532478\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 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
−14.47501228207756, −13.83590528622677, −13.06961276571997, −12.79941728334616, −12.16936517024201, −11.71438020299970, −11.39978735823991, −10.63401220740581, −10.14399816299859, −9.744430277644600, −8.999373964785017, −8.786860332374279, −7.792576717024579, −7.431631353016906, −7.287408846203010, −6.408450740955194, −5.617820874223527, −5.278662543259586, −4.789922057192559, −3.849624339446439, −3.468716106697056, −2.865030548490203, −2.090147652168205, −1.212552563802807, −0.5560209434085716,
0.5560209434085716, 1.212552563802807, 2.090147652168205, 2.865030548490203, 3.468716106697056, 3.849624339446439, 4.789922057192559, 5.278662543259586, 5.617820874223527, 6.408450740955194, 7.287408846203010, 7.431631353016906, 7.792576717024579, 8.786860332374279, 8.999373964785017, 9.744430277644600, 10.14399816299859, 10.63401220740581, 11.39978735823991, 11.71438020299970, 12.16936517024201, 12.79941728334616, 13.06961276571997, 13.83590528622677, 14.47501228207756
