| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 13-s + 14-s + 16-s − 3·17-s − 2·19-s − 3·23-s − 5·25-s + 26-s − 28-s − 10·31-s − 32-s + 3·34-s + 8·37-s + 2·38-s − 6·41-s + 4·43-s + 3·46-s + 49-s + 5·50-s − 52-s + 56-s + 3·59-s − 8·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.458·19-s − 0.625·23-s − 25-s + 0.196·26-s − 0.188·28-s − 1.79·31-s − 0.176·32-s + 0.514·34-s + 1.31·37-s + 0.324·38-s − 0.937·41-s + 0.609·43-s + 0.442·46-s + 1/7·49-s + 0.707·50-s − 0.138·52-s + 0.133·56-s + 0.390·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−13.19221758474346, −12.74347535857280, −12.43823328713031, −11.78451364998888, −11.30289073129511, −11.00412360911862, −10.43864137717590, −9.876085986618535, −9.601161893168727, −9.060622802820894, −8.644249186496768, −8.059432362413902, −7.626083508102162, −7.152055294168196, −6.619270636878707, −6.123821287167448, −5.684431575683630, −5.081816457501267, −4.336530961167239, −3.884786331609670, −3.324481736980487, −2.548151486772977, −2.092617928649104, −1.580327616903165, −0.5985115162328990, 0,
0.5985115162328990, 1.580327616903165, 2.092617928649104, 2.548151486772977, 3.324481736980487, 3.884786331609670, 4.336530961167239, 5.081816457501267, 5.684431575683630, 6.123821287167448, 6.619270636878707, 7.152055294168196, 7.626083508102162, 8.059432362413902, 8.644249186496768, 9.060622802820894, 9.601161893168727, 9.876085986618535, 10.43864137717590, 11.00412360911862, 11.30289073129511, 11.78451364998888, 12.43823328713031, 12.74347535857280, 13.19221758474346
