| L(s) = 1 | − 3-s + 2·5-s + 7-s − 2·9-s − 5·11-s − 2·13-s − 2·15-s + 2·19-s − 21-s + 4·23-s − 25-s + 5·27-s + 3·29-s + 31-s + 5·33-s + 2·35-s − 37-s + 2·39-s − 6·41-s − 8·43-s − 4·45-s − 12·47-s − 6·49-s − 14·53-s − 10·55-s − 2·57-s + 3·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s − 2/3·9-s − 1.50·11-s − 0.554·13-s − 0.516·15-s + 0.458·19-s − 0.218·21-s + 0.834·23-s − 1/5·25-s + 0.962·27-s + 0.557·29-s + 0.179·31-s + 0.870·33-s + 0.338·35-s − 0.164·37-s + 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.596·45-s − 1.75·47-s − 6/7·49-s − 1.92·53-s − 1.34·55-s − 0.264·57-s + 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 383792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 383792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| 83 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - T + p T^{2} \) | 1.7.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.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−12.78589626722913, −12.24991449276253, −11.60586888707912, −11.40527293925509, −10.96037341606390, −10.34023612928511, −10.03465349215178, −9.753280764621279, −9.104786231744511, −8.494331407014280, −8.187995305570446, −7.740431697948710, −7.115487890313136, −6.618796024539292, −6.176872943319788, −5.635236384154489, −5.217575355714009, −4.856702988915364, −4.640638696270799, −3.481347552991476, −2.988023970732674, −2.723414951234073, −1.862812863158823, −1.587207258151070, −0.5907071961988957, 0,
0.5907071961988957, 1.587207258151070, 1.862812863158823, 2.723414951234073, 2.988023970732674, 3.481347552991476, 4.640638696270799, 4.856702988915364, 5.217575355714009, 5.635236384154489, 6.176872943319788, 6.618796024539292, 7.115487890313136, 7.740431697948710, 8.187995305570446, 8.494331407014280, 9.104786231744511, 9.753280764621279, 10.03465349215178, 10.34023612928511, 10.96037341606390, 11.40527293925509, 11.60586888707912, 12.24991449276253, 12.78589626722913
