| L(s) = 1 | + 3-s − 2·5-s + 9-s + 3·11-s − 2·13-s − 2·15-s + 17-s + 19-s − 7·23-s − 25-s + 27-s − 10·29-s − 6·31-s + 3·33-s + 8·37-s − 2·39-s + 6·41-s + 4·43-s − 2·45-s − 9·47-s + 51-s + 4·53-s − 6·55-s + 57-s − 6·59-s + 61-s + 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.904·11-s − 0.554·13-s − 0.516·15-s + 0.242·17-s + 0.229·19-s − 1.45·23-s − 1/5·25-s + 0.192·27-s − 1.85·29-s − 1.07·31-s + 0.522·33-s + 1.31·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s − 1.31·47-s + 0.140·51-s + 0.549·53-s − 0.809·55-s + 0.132·57-s − 0.781·59-s + 0.128·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.046023728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.046023728\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−13.18258330793196, −12.60232541112843, −12.28840308153470, −11.73437047311940, −11.28459307207376, −11.05682520126239, −10.12738889330883, −9.844142008833953, −9.284744202506514, −8.963239621095960, −8.342358654623104, −7.716908978250763, −7.475231800789117, −7.230191875505804, −6.127170587705373, −6.094131176784033, −5.280822973793151, −4.568350458619795, −3.990023359837766, −3.841628619315603, −3.188665948657935, −2.485348472711418, −1.852428430898066, −1.299371630033038, −0.2799327600343729,
0.2799327600343729, 1.299371630033038, 1.852428430898066, 2.485348472711418, 3.188665948657935, 3.841628619315603, 3.990023359837766, 4.568350458619795, 5.280822973793151, 6.094131176784033, 6.127170587705373, 7.230191875505804, 7.475231800789117, 7.716908978250763, 8.342358654623104, 8.963239621095960, 9.284744202506514, 9.844142008833953, 10.12738889330883, 11.05682520126239, 11.28459307207376, 11.73437047311940, 12.28840308153470, 12.60232541112843, 13.18258330793196
