| L(s) = 1 | − 2-s − 3·3-s + 4-s + 3·6-s − 4·7-s − 8-s + 6·9-s − 3·12-s + 4·14-s + 16-s − 6·18-s + 2·19-s + 12·21-s + 23-s + 3·24-s − 9·27-s − 4·28-s − 2·29-s + 5·31-s − 32-s + 6·36-s − 2·37-s − 2·38-s + 2·41-s − 12·42-s − 8·43-s − 46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 1.22·6-s − 1.51·7-s − 0.353·8-s + 2·9-s − 0.866·12-s + 1.06·14-s + 1/4·16-s − 1.41·18-s + 0.458·19-s + 2.61·21-s + 0.208·23-s + 0.612·24-s − 1.73·27-s − 0.755·28-s − 0.371·29-s + 0.898·31-s − 0.176·32-s + 36-s − 0.328·37-s − 0.324·38-s + 0.312·41-s − 1.85·42-s − 1.21·43-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.00483426413081, −12.86198231080524, −12.19762025688071, −11.78393086162253, −11.57734665971029, −10.83337028976689, −10.55443308189539, −10.03280817456192, −9.675828272907641, −9.350239990291100, −8.615898915464133, −8.077916452766245, −7.303035859595600, −6.993356291730439, −6.563346902162962, −6.123296428544103, −5.744116327116961, −5.133391251801037, −4.677561798957688, −3.841095893026680, −3.415537474937816, −2.705922410240038, −1.967474491396903, −1.114128800962069, −0.6180292567703500, 0,
0.6180292567703500, 1.114128800962069, 1.967474491396903, 2.705922410240038, 3.415537474937816, 3.841095893026680, 4.677561798957688, 5.133391251801037, 5.744116327116961, 6.123296428544103, 6.563346902162962, 6.993356291730439, 7.303035859595600, 8.077916452766245, 8.615898915464133, 9.350239990291100, 9.675828272907641, 10.03280817456192, 10.55443308189539, 10.83337028976689, 11.57734665971029, 11.78393086162253, 12.19762025688071, 12.86198231080524, 13.00483426413081
