| L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s − 4·11-s − 13-s + 14-s − 16-s − 2·17-s + 4·19-s − 4·22-s + 4·23-s − 26-s − 28-s − 6·29-s + 8·31-s + 5·32-s − 2·34-s + 2·37-s + 4·38-s − 2·41-s + 4·43-s + 4·44-s + 4·46-s − 12·47-s + 49-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 1.20·11-s − 0.277·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.852·22-s + 0.834·23-s − 0.196·26-s − 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.883·32-s − 0.342·34-s + 0.328·37-s + 0.648·38-s − 0.312·41-s + 0.609·43-s + 0.603·44-s + 0.589·46-s − 1.75·47-s + 1/7·49-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−15.86095113559569, −15.19388486337353, −14.76290606847525, −14.29248504174000, −13.58022117371590, −13.22735018011688, −12.84624594006897, −12.20417191674873, −11.43218240969709, −11.25529922306962, −10.24775478493962, −9.875251752305908, −9.213774903209144, −8.576994623616632, −8.032452424033314, −7.445121409096356, −6.722977981169934, −5.944299433512188, −5.319714249244781, −4.908697544279429, −4.379168736026353, −3.487146050011393, −2.906195188549689, −2.208460427809642, −1.010230044926527, 0,
1.010230044926527, 2.208460427809642, 2.906195188549689, 3.487146050011393, 4.379168736026353, 4.908697544279429, 5.319714249244781, 5.944299433512188, 6.722977981169934, 7.445121409096356, 8.032452424033314, 8.576994623616632, 9.213774903209144, 9.875251752305908, 10.24775478493962, 11.25529922306962, 11.43218240969709, 12.20417191674873, 12.84624594006897, 13.22735018011688, 13.58022117371590, 14.29248504174000, 14.76290606847525, 15.19388486337353, 15.86095113559569
