| L(s) = 1 | + 11-s − 4·13-s + 8·19-s − 4·23-s − 2·29-s − 4·31-s − 8·37-s + 2·41-s + 4·47-s − 7·49-s + 12·53-s − 12·59-s + 2·61-s + 12·67-s + 12·73-s − 12·79-s + 8·83-s − 6·89-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s − 1.10·13-s + 1.83·19-s − 0.834·23-s − 0.371·29-s − 0.718·31-s − 1.31·37-s + 0.312·41-s + 0.583·47-s − 49-s + 1.64·53-s − 1.56·59-s + 0.256·61-s + 1.46·67-s + 1.40·73-s − 1.35·79-s + 0.878·83-s − 0.635·89-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−15.13008342638712, −14.41514149168537, −13.92919027456351, −13.80867592312146, −12.78914826628283, −12.50448148430400, −11.88747235246187, −11.51541768184233, −10.93200051482715, −10.09414862788810, −9.894926508283138, −9.254007360674534, −8.782535456496525, −8.003793150165018, −7.415963051656072, −7.173685389673713, −6.385529094157349, −5.666684711363315, −5.201989890326458, −4.661071968887809, −3.759195791595482, −3.369150367736184, −2.488932393302795, −1.862547163759111, −0.9778439086862975, 0,
0.9778439086862975, 1.862547163759111, 2.488932393302795, 3.369150367736184, 3.759195791595482, 4.661071968887809, 5.201989890326458, 5.666684711363315, 6.385529094157349, 7.173685389673713, 7.415963051656072, 8.003793150165018, 8.782535456496525, 9.254007360674534, 9.894926508283138, 10.09414862788810, 10.93200051482715, 11.51541768184233, 11.88747235246187, 12.50448148430400, 12.78914826628283, 13.80867592312146, 13.92919027456351, 14.41514149168537, 15.13008342638712
