| L(s) = 1 | + 7-s − 6·11-s − 13-s + 4·19-s + 4·31-s + 4·37-s − 6·41-s − 10·43-s + 6·47-s + 49-s + 6·53-s − 10·61-s − 4·67-s + 6·71-s − 2·73-s − 6·77-s + 4·79-s + 6·83-s − 6·89-s − 91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.80·11-s − 0.277·13-s + 0.917·19-s + 0.718·31-s + 0.657·37-s − 0.937·41-s − 1.52·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s − 1.28·61-s − 0.488·67-s + 0.712·71-s − 0.234·73-s − 0.683·77-s + 0.450·79-s + 0.658·83-s − 0.635·89-s − 0.104·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−12.95483932007743, −12.26284346074162, −11.98012170633207, −11.49221782506544, −11.00437429394434, −10.46045690631870, −10.17105636094454, −9.808105407051791, −9.117762003744353, −8.710427289308004, −8.053304672336334, −7.836843203188484, −7.420098915730158, −6.852987574319820, −6.282902936324757, −5.702185602491816, −5.146911545323934, −4.996139683075126, −4.388106448889560, −3.682632849366434, −3.073005759725179, −2.668718076193147, −2.127896424881506, −1.443937694675330, −0.7023612338512182, 0,
0.7023612338512182, 1.443937694675330, 2.127896424881506, 2.668718076193147, 3.073005759725179, 3.682632849366434, 4.388106448889560, 4.996139683075126, 5.146911545323934, 5.702185602491816, 6.282902936324757, 6.852987574319820, 7.420098915730158, 7.836843203188484, 8.053304672336334, 8.710427289308004, 9.117762003744353, 9.808105407051791, 10.17105636094454, 10.46045690631870, 11.00437429394434, 11.49221782506544, 11.98012170633207, 12.26284346074162, 12.95483932007743
