| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 3·7-s + 8-s − 2·9-s + 10-s + 11-s − 12-s + 13-s + 3·14-s − 15-s + 16-s − 2·18-s − 2·19-s + 20-s − 3·21-s + 22-s + 23-s − 24-s + 25-s + 26-s + 5·27-s + 3·28-s + 5·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.471·18-s − 0.458·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s + 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.962·27-s + 0.566·28-s + 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.278889531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.278889531\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 419 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.46189107865336, −14.19327550105202, −13.64036966179978, −13.19003692864052, −12.34909499003346, −12.13445969463762, −11.51160923942202, −11.06422244328071, −10.72526571855722, −10.12405684161092, −9.397240006446726, −8.697767649023274, −8.273345095116246, −7.733385535270783, −6.916249581998428, −6.382932503217888, −5.955995554976689, −5.371416281932374, −4.748776468552717, −4.486647544384212, −3.569411335335679, −2.840621229366577, −2.203340128065811, −1.442750307607615, −0.7042049388330634,
0.7042049388330634, 1.442750307607615, 2.203340128065811, 2.840621229366577, 3.569411335335679, 4.486647544384212, 4.748776468552717, 5.371416281932374, 5.955995554976689, 6.382932503217888, 6.916249581998428, 7.733385535270783, 8.273345095116246, 8.697767649023274, 9.397240006446726, 10.12405684161092, 10.72526571855722, 11.06422244328071, 11.51160923942202, 12.13445969463762, 12.34909499003346, 13.19003692864052, 13.64036966179978, 14.19327550105202, 14.46189107865336
