| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 6·13-s + 16-s − 2·17-s + 2·19-s − 20-s + 6·23-s + 25-s − 6·26-s − 6·29-s + 4·31-s + 32-s − 2·34-s − 8·37-s + 2·38-s − 40-s − 8·41-s + 43-s + 6·46-s + 6·47-s − 7·49-s + 50-s − 6·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 1.66·13-s + 1/4·16-s − 0.485·17-s + 0.458·19-s − 0.223·20-s + 1.25·23-s + 1/5·25-s − 1.17·26-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s − 1.31·37-s + 0.324·38-s − 0.158·40-s − 1.24·41-s + 0.152·43-s + 0.884·46-s + 0.875·47-s − 49-s + 0.141·50-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 43 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 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.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.898148324491716194387804251591, −7.21275330548561051655163294535, −6.80572084166079788779902646709, −5.70786839439801811490628172082, −4.94710770953213782526752401201, −4.47821731239985872238841626614, −3.37723319216182948624501022676, −2.73058443180740872249780426335, −1.63110901992609325411383107007, 0,
1.63110901992609325411383107007, 2.73058443180740872249780426335, 3.37723319216182948624501022676, 4.47821731239985872238841626614, 4.94710770953213782526752401201, 5.70786839439801811490628172082, 6.80572084166079788779902646709, 7.21275330548561051655163294535, 7.898148324491716194387804251591
