| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 11-s − 2·12-s + 13-s + 16-s + 3·17-s − 18-s − 8·19-s + 22-s − 6·23-s + 2·24-s − 26-s + 4·27-s − 9·29-s + 4·31-s − 32-s + 2·33-s − 3·34-s + 36-s − 2·37-s + 8·38-s − 2·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.577·12-s + 0.277·13-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.83·19-s + 0.213·22-s − 1.25·23-s + 0.408·24-s − 0.196·26-s + 0.769·27-s − 1.67·29-s + 0.718·31-s − 0.176·32-s + 0.348·33-s − 0.514·34-s + 1/6·36-s − 0.328·37-s + 1.29·38-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.56546822358531, −12.32378027980701, −11.68108836023734, −11.36674655217460, −10.97274622852743, −10.50715634876107, −10.17406319032421, −9.721649900749567, −9.196141156633838, −8.527850670693098, −8.265287886328860, −7.803470288976919, −7.197838708223247, −6.700433164190904, −6.190424657575055, −5.926450029469518, −5.458344081890613, −4.857589470947570, −4.327447570082198, −3.712179351133120, −3.212293748519267, −2.296620651731405, −2.009437889628518, −1.262045429999355, −0.5109441091827911, 0,
0.5109441091827911, 1.262045429999355, 2.009437889628518, 2.296620651731405, 3.212293748519267, 3.712179351133120, 4.327447570082198, 4.857589470947570, 5.458344081890613, 5.926450029469518, 6.190424657575055, 6.700433164190904, 7.197838708223247, 7.803470288976919, 8.265287886328860, 8.527850670693098, 9.196141156633838, 9.721649900749567, 10.17406319032421, 10.50715634876107, 10.97274622852743, 11.36674655217460, 11.68108836023734, 12.32378027980701, 12.56546822358531
