| L(s) = 1 | − 2·2-s + 2·4-s − 3·7-s − 3·9-s + 4·11-s − 2·13-s + 6·14-s − 4·16-s − 8·17-s + 6·18-s − 3·19-s − 8·22-s − 3·23-s − 5·25-s + 4·26-s − 6·28-s + 7·29-s − 10·31-s + 8·32-s + 16·34-s − 6·36-s + 7·37-s + 6·38-s + 9·41-s + 43-s + 8·44-s + 6·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.13·7-s − 9-s + 1.20·11-s − 0.554·13-s + 1.60·14-s − 16-s − 1.94·17-s + 1.41·18-s − 0.688·19-s − 1.70·22-s − 0.625·23-s − 25-s + 0.784·26-s − 1.13·28-s + 1.29·29-s − 1.79·31-s + 1.41·32-s + 2.74·34-s − 36-s + 1.15·37-s + 0.973·38-s + 1.40·41-s + 0.152·43-s + 1.20·44-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 197 | \( 1 + T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 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
−11.63768790626971835185556947749, −10.87939044119722991367373344739, −9.670505384884660391398822501321, −9.141663261808505731463150219631, −8.282753935470816342481418791481, −6.92472041340395206972505718477, −6.17591940844279569664203110179, −4.15785519193072071590213469433, −2.33277116067346011831864956690, 0,
2.33277116067346011831864956690, 4.15785519193072071590213469433, 6.17591940844279569664203110179, 6.92472041340395206972505718477, 8.282753935470816342481418791481, 9.141663261808505731463150219631, 9.670505384884660391398822501321, 10.87939044119722991367373344739, 11.63768790626971835185556947749
