| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 3·9-s − 2·10-s + 13-s + 16-s + 6·17-s − 3·18-s + 8·19-s − 2·20-s − 25-s + 26-s − 10·29-s + 32-s + 6·34-s − 3·36-s + 6·37-s + 8·38-s − 2·40-s − 4·43-s + 6·45-s + 8·47-s − 7·49-s − 50-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 9-s − 0.632·10-s + 0.277·13-s + 1/4·16-s + 1.45·17-s − 0.707·18-s + 1.83·19-s − 0.447·20-s − 1/5·25-s + 0.196·26-s − 1.85·29-s + 0.176·32-s + 1.02·34-s − 1/2·36-s + 0.986·37-s + 1.29·38-s − 0.316·40-s − 0.609·43-s + 0.894·45-s + 1.16·47-s − 49-s − 0.141·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43706 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43706 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 41 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.95309047566019, −14.43046647653812, −13.83226738704182, −13.60661421307372, −12.80916728274620, −12.27293644068948, −11.75144274175832, −11.54436131630804, −11.01057530462308, −10.39404731901553, −9.567821710514861, −9.304164850240115, −8.411581684601067, −7.825396394108154, −7.547476585821267, −7.039123952191037, −5.927718387899477, −5.780652874572898, −5.228543317657359, −4.431261382466863, −3.805222143591856, −3.081628863161963, −3.060534158580915, −1.808116061355712, −1.024841992488408, 0,
1.024841992488408, 1.808116061355712, 3.060534158580915, 3.081628863161963, 3.805222143591856, 4.431261382466863, 5.228543317657359, 5.780652874572898, 5.927718387899477, 7.039123952191037, 7.547476585821267, 7.825396394108154, 8.411581684601067, 9.304164850240115, 9.567821710514861, 10.39404731901553, 11.01057530462308, 11.54436131630804, 11.75144274175832, 12.27293644068948, 12.80916728274620, 13.60661421307372, 13.83226738704182, 14.43046647653812, 14.95309047566019
