| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s + 4·11-s − 13-s + 16-s + 6·17-s + 4·19-s + 2·20-s + 4·22-s − 25-s − 26-s + 4·29-s − 4·31-s + 32-s + 6·34-s + 12·37-s + 4·38-s + 2·40-s + 12·41-s − 8·43-s + 4·44-s + 2·47-s − 50-s − 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s + 1.20·11-s − 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.447·20-s + 0.852·22-s − 1/5·25-s − 0.196·26-s + 0.742·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s + 1.97·37-s + 0.648·38-s + 0.316·40-s + 1.87·41-s − 1.21·43-s + 0.603·44-s + 0.291·47-s − 0.141·50-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.231567458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.231567458\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 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
−16.33432873309609, −16.07121141879274, −14.90050788801329, −14.74817845297972, −14.13243958492821, −13.68995603738613, −13.14795225139383, −12.27485010130774, −12.10190443101204, −11.36908330946999, −10.71756273219436, −9.955300210857007, −9.469715606973959, −9.095444527963656, −7.837681437470483, −7.617013519920214, −6.633362569928499, −6.054428568686478, −5.667545527416671, −4.842966792177620, −4.163204272336281, −3.321162346053718, −2.714072920284594, −1.668067351069134, −1.042944646143161,
1.042944646143161, 1.668067351069134, 2.714072920284594, 3.321162346053718, 4.163204272336281, 4.842966792177620, 5.667545527416671, 6.054428568686478, 6.633362569928499, 7.617013519920214, 7.837681437470483, 9.095444527963656, 9.469715606973959, 9.955300210857007, 10.71756273219436, 11.36908330946999, 12.10190443101204, 12.27485010130774, 13.14795225139383, 13.68995603738613, 14.13243958492821, 14.74817845297972, 14.90050788801329, 16.07121141879274, 16.33432873309609
