| L(s) = 1 | − 3·3-s + 6·9-s − 11-s + 13-s − 2·17-s + 2·19-s − 6·23-s − 5·25-s − 9·27-s − 29-s − 4·31-s + 3·33-s − 2·37-s − 3·39-s + 2·41-s − 8·43-s + 6·47-s + 6·51-s − 8·53-s − 6·57-s + 59-s + 5·61-s − 9·67-s + 18·69-s − 4·71-s + 10·73-s + 15·75-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s − 0.301·11-s + 0.277·13-s − 0.485·17-s + 0.458·19-s − 1.25·23-s − 25-s − 1.73·27-s − 0.185·29-s − 0.718·31-s + 0.522·33-s − 0.328·37-s − 0.480·39-s + 0.312·41-s − 1.21·43-s + 0.875·47-s + 0.840·51-s − 1.09·53-s − 0.794·57-s + 0.130·59-s + 0.640·61-s − 1.09·67-s + 2.16·69-s − 0.474·71-s + 1.17·73-s + 1.73·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4510285081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4510285081\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−15.94888555260058, −15.63339431471510, −14.93090370097975, −14.06796917890186, −13.60388630667147, −12.86713681788415, −12.52192719989457, −11.74440136678757, −11.51736212159785, −10.96361890829384, −10.24281845099955, −9.987398343553012, −9.204132561945143, −8.405901331017475, −7.628827704074642, −7.149272498876092, −6.387283242403565, −5.903108903125986, −5.455826828915308, −4.735776896421051, −4.134298936463253, −3.393871030626738, −2.152081002731124, −1.428404996982182, −0.3247717880786707,
0.3247717880786707, 1.428404996982182, 2.152081002731124, 3.393871030626738, 4.134298936463253, 4.735776896421051, 5.455826828915308, 5.903108903125986, 6.387283242403565, 7.149272498876092, 7.628827704074642, 8.405901331017475, 9.204132561945143, 9.987398343553012, 10.24281845099955, 10.96361890829384, 11.51736212159785, 11.74440136678757, 12.52192719989457, 12.86713681788415, 13.60388630667147, 14.06796917890186, 14.93090370097975, 15.63339431471510, 15.94888555260058
