| L(s) = 1 | − 3·5-s − 2·7-s − 13-s − 2·19-s + 3·23-s + 4·25-s − 6·29-s − 31-s + 6·35-s − 7·37-s + 6·41-s − 8·43-s − 12·47-s − 3·49-s + 6·53-s − 9·59-s − 2·61-s + 3·65-s − 7·67-s + 3·71-s − 8·73-s + 4·79-s − 12·83-s + 15·89-s + 2·91-s + 6·95-s − 13·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.755·7-s − 0.277·13-s − 0.458·19-s + 0.625·23-s + 4/5·25-s − 1.11·29-s − 0.179·31-s + 1.01·35-s − 1.15·37-s + 0.937·41-s − 1.21·43-s − 1.75·47-s − 3/7·49-s + 0.824·53-s − 1.17·59-s − 0.256·61-s + 0.372·65-s − 0.855·67-s + 0.356·71-s − 0.936·73-s + 0.450·79-s − 1.31·83-s + 1.58·89-s + 0.209·91-s + 0.615·95-s − 1.31·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56628 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56628 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−14.95527539463591, −14.62868329887374, −13.77120035139687, −13.30871668750768, −12.72222743207289, −12.43406970657944, −11.81202867693304, −11.33353417716574, −10.95640026129164, −10.24942685736327, −9.815224420094520, −9.040345334111940, −8.799993896020893, −7.907113771795346, −7.747314766533202, −6.925081096959185, −6.696105996851125, −5.896997350539964, −5.228633547290170, −4.590378323243598, −4.050916613320807, −3.339816745746097, −3.119688236507481, −2.113270921248683, −1.282284422516678, 0, 0,
1.282284422516678, 2.113270921248683, 3.119688236507481, 3.339816745746097, 4.050916613320807, 4.590378323243598, 5.228633547290170, 5.896997350539964, 6.696105996851125, 6.925081096959185, 7.747314766533202, 7.907113771795346, 8.799993896020893, 9.040345334111940, 9.815224420094520, 10.24942685736327, 10.95640026129164, 11.33353417716574, 11.81202867693304, 12.43406970657944, 12.72222743207289, 13.30871668750768, 13.77120035139687, 14.62868329887374, 14.95527539463591
