| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s − 5·11-s − 12-s − 2·13-s + 14-s − 15-s + 16-s + 18-s − 2·19-s + 20-s − 21-s − 5·22-s − 4·23-s − 24-s − 4·25-s − 2·26-s − 27-s + 28-s + 9·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.50·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s + 0.223·20-s − 0.218·21-s − 1.06·22-s − 0.834·23-s − 0.204·24-s − 4/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.472577466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.472577466\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−16.25715437447140, −15.83674274743958, −15.02496962920677, −14.90463387899870, −13.82810310736521, −13.56163818784243, −13.03680687172514, −12.38679703289874, −11.66133977046801, −11.54175103936727, −10.42220818842518, −10.16304350895826, −9.807083288047245, −8.447924161216223, −8.114221239531532, −7.442612523394064, −6.500605477510542, −6.213215803356122, −5.330415167007685, −4.881958770561925, −4.397652356021472, −3.309850981932637, −2.477561026446859, −1.927581902235528, −0.6360052246322109,
0.6360052246322109, 1.927581902235528, 2.477561026446859, 3.309850981932637, 4.397652356021472, 4.881958770561925, 5.330415167007685, 6.213215803356122, 6.500605477510542, 7.442612523394064, 8.114221239531532, 8.447924161216223, 9.807083288047245, 10.16304350895826, 10.42220818842518, 11.54175103936727, 11.66133977046801, 12.38679703289874, 13.03680687172514, 13.56163818784243, 13.82810310736521, 14.90463387899870, 15.02496962920677, 15.83674274743958, 16.25715437447140
