| L(s) = 1 | − 3·9-s + 2·11-s + 2·13-s + 6·17-s − 8·19-s + 8·23-s − 5·25-s − 4·29-s − 8·31-s + 8·37-s + 6·41-s + 8·43-s − 8·47-s − 7·49-s − 10·53-s − 2·59-s − 10·61-s + 14·67-s − 71-s + 14·73-s + 9·81-s − 12·83-s + 6·89-s − 2·97-s − 6·99-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 9-s + 0.603·11-s + 0.554·13-s + 1.45·17-s − 1.83·19-s + 1.66·23-s − 25-s − 0.742·29-s − 1.43·31-s + 1.31·37-s + 0.937·41-s + 1.21·43-s − 1.16·47-s − 49-s − 1.37·53-s − 0.260·59-s − 1.28·61-s + 1.71·67-s − 0.118·71-s + 1.63·73-s + 81-s − 1.31·83-s + 0.635·89-s − 0.203·97-s − 0.603·99-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 71 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−16.25214986463655, −15.38801736373132, −14.81500317271494, −14.47683770453931, −14.09834840014361, −13.14982416492604, −12.80493050730535, −12.31625660830797, −11.44784496271310, −10.95347136997035, −10.88375382452487, −9.657091492786821, −9.367026833850282, −8.779406238594252, −8.005192620535905, −7.726525840902767, −6.741907189893179, −6.176771525364178, −5.703026470459377, −5.013731013603538, −4.129699153238517, −3.546791816062859, −2.868999190292452, −1.977097414351048, −1.115790467446046, 0,
1.115790467446046, 1.977097414351048, 2.868999190292452, 3.546791816062859, 4.129699153238517, 5.013731013603538, 5.703026470459377, 6.176771525364178, 6.741907189893179, 7.726525840902767, 8.005192620535905, 8.779406238594252, 9.367026833850282, 9.657091492786821, 10.88375382452487, 10.95347136997035, 11.44784496271310, 12.31625660830797, 12.80493050730535, 13.14982416492604, 14.09834840014361, 14.47683770453931, 14.81500317271494, 15.38801736373132, 16.25214986463655
