| L(s) = 1 | + 2-s + 3-s − 4-s + 3·5-s + 6-s + 7-s − 3·8-s + 9-s + 3·10-s − 11-s − 12-s − 13-s + 14-s + 3·15-s − 16-s + 18-s + 7·19-s − 3·20-s + 21-s − 22-s − 23-s − 3·24-s + 4·25-s − 26-s + 27-s − 28-s + 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 1.34·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.948·10-s − 0.301·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.774·15-s − 1/4·16-s + 0.235·18-s + 1.60·19-s − 0.670·20-s + 0.218·21-s − 0.213·22-s − 0.208·23-s − 0.612·24-s + 4/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123981 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123981 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 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 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−13.78390790459310, −13.32619038367196, −13.11856984408378, −12.53140800838196, −11.88839271393395, −11.59112508775166, −10.85113639538301, −10.05531825491839, −9.907434604751052, −9.429899426685540, −9.055635290254499, −8.295262437196801, −8.032128287858728, −7.359932633106354, −6.564682085461324, −6.298826704505978, −5.441708168721769, −5.265252958596704, −4.770323584214517, −4.153239794565256, −3.318783045006577, −3.041459402108439, −2.379693567795888, −1.665795226974087, −1.120069213906982, 0,
1.120069213906982, 1.665795226974087, 2.379693567795888, 3.041459402108439, 3.318783045006577, 4.153239794565256, 4.770323584214517, 5.265252958596704, 5.441708168721769, 6.298826704505978, 6.564682085461324, 7.359932633106354, 8.032128287858728, 8.295262437196801, 9.055635290254499, 9.429899426685540, 9.907434604751052, 10.05531825491839, 10.85113639538301, 11.59112508775166, 11.88839271393395, 12.53140800838196, 13.11856984408378, 13.32619038367196, 13.78390790459310
