| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 3·9-s − 2·10-s + 5·11-s + 2·13-s + 16-s + 4·17-s + 3·18-s − 8·19-s + 2·20-s − 5·22-s − 25-s − 2·26-s + 3·29-s − 32-s − 4·34-s − 3·36-s − 3·37-s + 8·38-s − 2·40-s + 2·41-s − 8·43-s + 5·44-s − 6·45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 9-s − 0.632·10-s + 1.50·11-s + 0.554·13-s + 1/4·16-s + 0.970·17-s + 0.707·18-s − 1.83·19-s + 0.447·20-s − 1.06·22-s − 1/5·25-s − 0.392·26-s + 0.557·29-s − 0.176·32-s − 0.685·34-s − 1/2·36-s − 0.493·37-s + 1.29·38-s − 0.316·40-s + 0.312·41-s − 1.21·43-s + 0.753·44-s − 0.894·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.772250764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.772250764\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 11 T + p T^{2} \) | 1.71.al |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 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
−8.220701567923354224541276128378, −7.03270191722903092083144502088, −6.46040163731463222784437976129, −5.96893621998516061171982111211, −5.35094116344613477775713597842, −4.15259553690520012968954992591, −3.45988419950193376300645464659, −2.42097322724825255638829502298, −1.73519115986432019918067141000, −0.75380318991393529487366787938,
0.75380318991393529487366787938, 1.73519115986432019918067141000, 2.42097322724825255638829502298, 3.45988419950193376300645464659, 4.15259553690520012968954992591, 5.35094116344613477775713597842, 5.96893621998516061171982111211, 6.46040163731463222784437976129, 7.03270191722903092083144502088, 8.220701567923354224541276128378
