L(s) = 1 | + 5-s − 2·7-s + 5·13-s − 17-s − 19-s − 7·23-s − 4·25-s + 6·29-s − 4·31-s − 2·35-s + 10·37-s + 9·41-s + 43-s − 12·47-s − 3·49-s + 12·53-s + 6·59-s − 2·61-s + 5·65-s − 4·67-s − 8·71-s − 6·79-s − 4·83-s − 85-s − 2·89-s − 10·91-s − 95-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s + 1.38·13-s − 0.242·17-s − 0.229·19-s − 1.45·23-s − 4/5·25-s + 1.11·29-s − 0.718·31-s − 0.338·35-s + 1.64·37-s + 1.40·41-s + 0.152·43-s − 1.75·47-s − 3/7·49-s + 1.64·53-s + 0.781·59-s − 0.256·61-s + 0.620·65-s − 0.488·67-s − 0.949·71-s − 0.675·79-s − 0.439·83-s − 0.108·85-s − 0.211·89-s − 1.04·91-s − 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−12.87074657367544, −12.79320284187444, −11.88014761323964, −11.51553387671655, −11.25604014739953, −10.39833731667759, −10.26833819971692, −9.737946148566706, −9.308027755419579, −8.738941245389600, −8.422076384074418, −7.774860236898491, −7.447137323186974, −6.587008564764510, −6.340238120527611, −5.866081852379237, −5.668374158291025, −4.705480490291384, −4.260091844790690, −3.764514427833090, −3.289999734455752, −2.590213763855256, −2.117081823713344, −1.445062664050849, −0.7864376196382707, 0,
0.7864376196382707, 1.445062664050849, 2.117081823713344, 2.590213763855256, 3.289999734455752, 3.764514427833090, 4.260091844790690, 4.705480490291384, 5.668374158291025, 5.866081852379237, 6.340238120527611, 6.587008564764510, 7.447137323186974, 7.774860236898491, 8.422076384074418, 8.738941245389600, 9.308027755419579, 9.737946148566706, 10.26833819971692, 10.39833731667759, 11.25604014739953, 11.51553387671655, 11.88014761323964, 12.79320284187444, 12.87074657367544