| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 4·11-s − 13-s + 2·14-s + 16-s + 17-s + 4·22-s + 23-s − 5·25-s + 26-s − 2·28-s + 6·29-s + 6·31-s − 32-s − 34-s + 4·37-s + 5·41-s + 11·43-s − 4·44-s − 46-s − 10·47-s − 3·49-s + 5·50-s − 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 1.20·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.852·22-s + 0.208·23-s − 25-s + 0.196·26-s − 0.377·28-s + 1.11·29-s + 1.07·31-s − 0.176·32-s − 0.171·34-s + 0.657·37-s + 0.780·41-s + 1.67·43-s − 0.603·44-s − 0.147·46-s − 1.45·47-s − 3/7·49-s + 0.707·50-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 7 T + p T^{2} \) | 1.53.ah |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−13.41172784597058, −13.15158530443092, −12.54342079532298, −12.21357224172407, −11.58692144652646, −11.13607936033321, −10.62831217426485, −10.04628868243876, −9.781076138160873, −9.491033689188010, −8.607884315140498, −8.277335398876485, −7.874008132991648, −7.211927088648572, −6.905636306518880, −6.170945014871594, −5.798634271452583, −5.257150171668507, −4.527855540811910, −4.013096863731383, −3.210302787408799, −2.579876734259561, −2.477733623759265, −1.396378288678672, −0.7020160790638200, 0,
0.7020160790638200, 1.396378288678672, 2.477733623759265, 2.579876734259561, 3.210302787408799, 4.013096863731383, 4.527855540811910, 5.257150171668507, 5.798634271452583, 6.170945014871594, 6.905636306518880, 7.211927088648572, 7.874008132991648, 8.277335398876485, 8.607884315140498, 9.491033689188010, 9.781076138160873, 10.04628868243876, 10.62831217426485, 11.13607936033321, 11.58692144652646, 12.21357224172407, 12.54342079532298, 13.15158530443092, 13.41172784597058
