| L(s) = 1 | + 3-s − 5-s − 3·7-s + 9-s + 3·11-s − 15-s − 3·19-s − 3·21-s + 4·23-s + 25-s + 27-s − 4·29-s − 6·31-s + 3·33-s + 3·35-s + 9·37-s − 10·41-s + 10·43-s − 45-s + 3·47-s + 2·49-s + 9·53-s − 3·55-s − 3·57-s − 12·59-s − 6·61-s − 3·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s + 0.904·11-s − 0.258·15-s − 0.688·19-s − 0.654·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s − 1.07·31-s + 0.522·33-s + 0.507·35-s + 1.47·37-s − 1.56·41-s + 1.52·43-s − 0.149·45-s + 0.437·47-s + 2/7·49-s + 1.23·53-s − 0.404·55-s − 0.397·57-s − 1.56·59-s − 0.768·61-s − 0.377·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 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
−14.95332197292897, −14.61331056704962, −14.04302547080939, −13.37061039723334, −12.97192252067025, −12.52229435921915, −12.06461202600954, −11.21866311372910, −10.99480870182577, −10.16200615228349, −9.679058188448393, −9.074934621831141, −8.879979366055909, −8.135666916666670, −7.368142308430252, −7.098936729349219, −6.378046157065600, −5.950572569282563, −5.118302192137108, −4.253037406253480, −3.890498838506729, −3.261973792944471, −2.687812675586317, −1.862221125230959, −0.9673094416971007, 0,
0.9673094416971007, 1.862221125230959, 2.687812675586317, 3.261973792944471, 3.890498838506729, 4.253037406253480, 5.118302192137108, 5.950572569282563, 6.378046157065600, 7.098936729349219, 7.368142308430252, 8.135666916666670, 8.879979366055909, 9.074934621831141, 9.679058188448393, 10.16200615228349, 10.99480870182577, 11.21866311372910, 12.06461202600954, 12.52229435921915, 12.97192252067025, 13.37061039723334, 14.04302547080939, 14.61331056704962, 14.95332197292897
