| L(s) = 1 | − 3·5-s − 4·7-s + 11-s + 2·13-s + 8·17-s − 6·19-s − 5·23-s + 4·25-s + 4·29-s + 31-s + 12·35-s − 3·37-s + 6·41-s + 6·43-s + 12·47-s + 9·49-s − 6·53-s − 3·55-s + 3·59-s − 6·65-s − 11·67-s + 5·71-s − 10·73-s − 4·77-s − 2·79-s − 2·83-s − 24·85-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.51·7-s + 0.301·11-s + 0.554·13-s + 1.94·17-s − 1.37·19-s − 1.04·23-s + 4/5·25-s + 0.742·29-s + 0.179·31-s + 2.02·35-s − 0.493·37-s + 0.937·41-s + 0.914·43-s + 1.75·47-s + 9/7·49-s − 0.824·53-s − 0.404·55-s + 0.390·59-s − 0.744·65-s − 1.34·67-s + 0.593·71-s − 1.17·73-s − 0.455·77-s − 0.225·79-s − 0.219·83-s − 2.60·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−7.64917317542877670356044654074, −7.09189229634945910249225174837, −6.12575951092354248118947396597, −5.89202106517789488102139723477, −4.53287493758271029911779130814, −3.83733764673944861598118900337, −3.43778877144274756195405556734, −2.54702017160053200485681408944, −1.02281222444134172563248729402, 0,
1.02281222444134172563248729402, 2.54702017160053200485681408944, 3.43778877144274756195405556734, 3.83733764673944861598118900337, 4.53287493758271029911779130814, 5.89202106517789488102139723477, 6.12575951092354248118947396597, 7.09189229634945910249225174837, 7.64917317542877670356044654074
