| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s − 2·9-s − 2·12-s + 4·13-s − 4·16-s + 17-s + 4·18-s + 7·19-s − 5·23-s − 8·26-s + 5·27-s + 8·29-s + 4·31-s + 8·32-s − 2·34-s − 4·36-s − 37-s − 14·38-s − 4·39-s − 4·41-s − 2·43-s + 10·46-s − 12·47-s + 4·48-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s − 2/3·9-s − 0.577·12-s + 1.10·13-s − 16-s + 0.242·17-s + 0.942·18-s + 1.60·19-s − 1.04·23-s − 1.56·26-s + 0.962·27-s + 1.48·29-s + 0.718·31-s + 1.41·32-s − 0.342·34-s − 2/3·36-s − 0.164·37-s − 2.27·38-s − 0.640·39-s − 0.624·41-s − 0.304·43-s + 1.47·46-s − 1.75·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−14.78914340521171, −14.06953835343360, −13.84900768221296, −13.30123794774505, −12.50151166210327, −11.80965955286353, −11.58657910444428, −11.13624188407321, −10.49365132216563, −9.974824330572778, −9.728510746894006, −8.948120370137341, −8.390104708589225, −8.170857066654410, −7.575991447374349, −6.791669073994361, −6.349851168686921, −5.877446744665481, −5.013726072403165, −4.689275542443890, −3.562472790885838, −3.126467647054512, −2.214443836192780, −1.347796115844585, −0.8715364585234548, 0,
0.8715364585234548, 1.347796115844585, 2.214443836192780, 3.126467647054512, 3.562472790885838, 4.689275542443890, 5.013726072403165, 5.877446744665481, 6.349851168686921, 6.791669073994361, 7.575991447374349, 8.170857066654410, 8.390104708589225, 8.948120370137341, 9.728510746894006, 9.974824330572778, 10.49365132216563, 11.13624188407321, 11.58657910444428, 11.80965955286353, 12.50151166210327, 13.30123794774505, 13.84900768221296, 14.06953835343360, 14.78914340521171
