| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 11-s − 14-s + 16-s + 17-s + 19-s − 22-s − 8·23-s − 5·25-s + 28-s − 6·29-s + 2·31-s − 32-s − 34-s − 5·37-s − 38-s + 2·41-s − 11·43-s + 44-s + 8·46-s + 47-s + 49-s + 5·50-s + 9·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.301·11-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.229·19-s − 0.213·22-s − 1.66·23-s − 25-s + 0.188·28-s − 1.11·29-s + 0.359·31-s − 0.176·32-s − 0.171·34-s − 0.821·37-s − 0.162·38-s + 0.312·41-s − 1.67·43-s + 0.150·44-s + 1.17·46-s + 0.145·47-s + 1/7·49-s + 0.707·50-s + 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.94596632002436, −12.19786501193394, −12.06279390315018, −11.56924410815962, −11.25476627332276, −10.53507423929440, −10.26115306380796, −9.776619100015033, −9.425643835625123, −8.814813291903636, −8.426354431742164, −7.989926011332661, −7.468866869732832, −7.224339194138982, −6.482418588498718, −6.076412834248302, −5.573421979201660, −5.187694799432001, −4.306720620164131, −4.023700815380686, −3.415488063669130, −2.795205478533021, −2.143381674326480, −1.592258357324414, −1.268993665328184, 0, 0,
1.268993665328184, 1.592258357324414, 2.143381674326480, 2.795205478533021, 3.415488063669130, 4.023700815380686, 4.306720620164131, 5.187694799432001, 5.573421979201660, 6.076412834248302, 6.482418588498718, 7.224339194138982, 7.468866869732832, 7.989926011332661, 8.426354431742164, 8.814813291903636, 9.425643835625123, 9.776619100015033, 10.26115306380796, 10.53507423929440, 11.25476627332276, 11.56924410815962, 12.06279390315018, 12.19786501193394, 12.94596632002436
