| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 11-s + 4·13-s − 15-s − 4·19-s + 2·21-s + 5·23-s − 4·25-s − 27-s − 6·29-s + 5·31-s − 33-s − 2·35-s − 2·37-s − 4·39-s + 2·41-s − 10·43-s + 45-s − 47-s − 3·49-s − 6·53-s + 55-s + 4·57-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.258·15-s − 0.917·19-s + 0.436·21-s + 1.04·23-s − 4/5·25-s − 0.192·27-s − 1.11·29-s + 0.898·31-s − 0.174·33-s − 0.338·35-s − 0.328·37-s − 0.640·39-s + 0.312·41-s − 1.52·43-s + 0.149·45-s − 0.145·47-s − 3/7·49-s − 0.824·53-s + 0.134·55-s + 0.529·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−13.43064454246191, −13.09988552378040, −12.69268349649145, −12.19248116963317, −11.54154340058965, −11.20944005487299, −10.71579933011624, −10.29681463260146, −9.611561521750461, −9.422451823208980, −8.822882650503350, −8.208690928820611, −7.826536850789648, −6.918177440006810, −6.580401524734150, −6.323787233773521, −5.652706192085485, −5.291550246943550, −4.531747525700719, −4.026153574266461, −3.405965419274706, −2.971039015062239, −2.003368723673990, −1.591705304952850, −0.7744506103666518, 0,
0.7744506103666518, 1.591705304952850, 2.003368723673990, 2.971039015062239, 3.405965419274706, 4.026153574266461, 4.531747525700719, 5.291550246943550, 5.652706192085485, 6.323787233773521, 6.580401524734150, 6.918177440006810, 7.826536850789648, 8.208690928820611, 8.822882650503350, 9.422451823208980, 9.611561521750461, 10.29681463260146, 10.71579933011624, 11.20944005487299, 11.54154340058965, 12.19248116963317, 12.69268349649145, 13.09988552378040, 13.43064454246191
