| L(s) = 1 | − 5-s − 2·11-s + 5·13-s + 2·17-s − 2·19-s − 23-s − 4·25-s + 5·29-s − 7·37-s + 3·41-s + 11·43-s − 9·47-s − 10·53-s + 2·55-s + 12·59-s + 2·61-s − 5·65-s + 4·67-s − 6·71-s − 6·73-s + 12·79-s − 4·83-s − 2·85-s + 2·95-s + 3·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.603·11-s + 1.38·13-s + 0.485·17-s − 0.458·19-s − 0.208·23-s − 4/5·25-s + 0.928·29-s − 1.15·37-s + 0.468·41-s + 1.67·43-s − 1.31·47-s − 1.37·53-s + 0.269·55-s + 1.56·59-s + 0.256·61-s − 0.620·65-s + 0.488·67-s − 0.712·71-s − 0.702·73-s + 1.35·79-s − 0.439·83-s − 0.216·85-s + 0.205·95-s + 0.304·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−13.27205933413351, −13.22921246993835, −12.49097752409700, −12.10914998756429, −11.62195112506179, −10.96742443248423, −10.84871222786238, −10.17607881609190, −9.764936648245534, −9.136990410781783, −8.559518133717891, −8.223534312755227, −7.788324579942001, −7.275239724721950, −6.535969238630424, −6.247758774355259, −5.578776105640075, −5.181780175650521, −4.402076423303991, −3.983354587343949, −3.442662685729695, −2.899318332339019, −2.180605050840768, −1.499518069682134, −0.8120441102891865, 0,
0.8120441102891865, 1.499518069682134, 2.180605050840768, 2.899318332339019, 3.442662685729695, 3.983354587343949, 4.402076423303991, 5.181780175650521, 5.578776105640075, 6.247758774355259, 6.535969238630424, 7.275239724721950, 7.788324579942001, 8.223534312755227, 8.559518133717891, 9.136990410781783, 9.764936648245534, 10.17607881609190, 10.84871222786238, 10.96742443248423, 11.62195112506179, 12.10914998756429, 12.49097752409700, 13.22921246993835, 13.27205933413351
