| L(s) = 1 | − 7-s − 4·11-s − 13-s + 17-s + 5·19-s + 4·23-s − 3·29-s + 8·31-s − 4·37-s − 2·41-s + 7·47-s + 49-s − 9·53-s − 10·59-s − 13·61-s − 6·67-s + 8·71-s − 2·73-s + 4·77-s + 3·79-s + 12·83-s − 11·89-s + 91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.20·11-s − 0.277·13-s + 0.242·17-s + 1.14·19-s + 0.834·23-s − 0.557·29-s + 1.43·31-s − 0.657·37-s − 0.312·41-s + 1.02·47-s + 1/7·49-s − 1.23·53-s − 1.30·59-s − 1.66·61-s − 0.733·67-s + 0.949·71-s − 0.234·73-s + 0.455·77-s + 0.337·79-s + 1.31·83-s − 1.16·89-s + 0.104·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 302400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 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 |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.89998567641884, −12.34591742823535, −12.17027571533598, −11.55275284601351, −10.97198445813366, −10.67553732503362, −10.10079947956770, −9.796162995915276, −9.215732644121795, −8.872971120321944, −8.199967736599069, −7.693167952658937, −7.455414560903022, −6.932715338832568, −6.200506147795905, −5.951735940626490, −5.156818556396696, −4.976239608550866, −4.434071054505256, −3.610056888784149, −3.080934990314497, −2.843445620365698, −2.114341475860209, −1.403129920514508, −0.7188979767565290, 0,
0.7188979767565290, 1.403129920514508, 2.114341475860209, 2.843445620365698, 3.080934990314497, 3.610056888784149, 4.434071054505256, 4.976239608550866, 5.156818556396696, 5.951735940626490, 6.200506147795905, 6.932715338832568, 7.455414560903022, 7.693167952658937, 8.199967736599069, 8.872971120321944, 9.215732644121795, 9.796162995915276, 10.10079947956770, 10.67553732503362, 10.97198445813366, 11.55275284601351, 12.17027571533598, 12.34591742823535, 12.89998567641884
