| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s − 3·9-s − 3·11-s + 4·14-s + 16-s + 4·17-s + 3·18-s + 3·22-s − 3·23-s − 4·28-s + 6·29-s − 3·31-s − 32-s − 4·34-s − 3·36-s − 37-s + 5·41-s − 6·43-s − 3·44-s + 3·46-s + 8·47-s + 9·49-s + 2·53-s + 4·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 9-s − 0.904·11-s + 1.06·14-s + 1/4·16-s + 0.970·17-s + 0.707·18-s + 0.639·22-s − 0.625·23-s − 0.755·28-s + 1.11·29-s − 0.538·31-s − 0.176·32-s − 0.685·34-s − 1/2·36-s − 0.164·37-s + 0.780·41-s − 0.914·43-s − 0.452·44-s + 0.442·46-s + 1.16·47-s + 9/7·49-s + 0.274·53-s + 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−12.70880790186887, −12.39196243941391, −12.01000793327817, −11.52033372421050, −10.87064406832579, −10.47286428205190, −10.16036790124275, −9.687526637116165, −9.180778793087510, −8.879510377858654, −8.226161944235548, −7.834536662615258, −7.467360960417031, −6.772260405550860, −6.315923759341945, −6.011231361079880, −5.373764835796964, −5.089518521095226, −4.086495607433924, −3.599972439607484, −3.001405987621938, −2.723980433912554, −2.185432262538855, −1.266342033728577, −0.5349630104538323, 0,
0.5349630104538323, 1.266342033728577, 2.185432262538855, 2.723980433912554, 3.001405987621938, 3.599972439607484, 4.086495607433924, 5.089518521095226, 5.373764835796964, 6.011231361079880, 6.315923759341945, 6.772260405550860, 7.467360960417031, 7.834536662615258, 8.226161944235548, 8.879510377858654, 9.180778793087510, 9.687526637116165, 10.16036790124275, 10.47286428205190, 10.87064406832579, 11.52033372421050, 12.01000793327817, 12.39196243941391, 12.70880790186887
