| L(s) = 1 | + 2·5-s + 4·11-s + 13-s − 6·17-s + 8·23-s − 25-s + 10·29-s − 8·31-s + 6·37-s − 6·41-s − 4·43-s + 8·47-s − 6·53-s + 8·55-s − 8·59-s − 10·61-s + 2·65-s − 4·67-s − 8·71-s − 2·73-s − 8·79-s − 12·85-s + 18·89-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s + 0.277·13-s − 1.45·17-s + 1.66·23-s − 1/5·25-s + 1.85·29-s − 1.43·31-s + 0.986·37-s − 0.937·41-s − 0.609·43-s + 1.16·47-s − 0.824·53-s + 1.07·55-s − 1.04·59-s − 1.28·61-s + 0.248·65-s − 0.488·67-s − 0.949·71-s − 0.234·73-s − 0.900·79-s − 1.30·85-s + 1.90·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 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
−13.98741792795436, −13.61327244201949, −13.17963442506065, −12.75790421968633, −12.07472850506848, −11.63355173950998, −11.11709670620690, −10.59263957106223, −10.24066702817719, −9.374187292986391, −9.113980958951052, −8.881343585619591, −8.156608986183838, −7.414093605988840, −6.844168822352281, −6.438481136285696, −6.063162656147895, −5.378756500233997, −4.643335302678469, −4.393993539743055, −3.526453697726948, −2.942964096170502, −2.286358125449646, −1.550004641744986, −1.122576431548871, 0,
1.122576431548871, 1.550004641744986, 2.286358125449646, 2.942964096170502, 3.526453697726948, 4.393993539743055, 4.643335302678469, 5.378756500233997, 6.063162656147895, 6.438481136285696, 6.844168822352281, 7.414093605988840, 8.156608986183838, 8.881343585619591, 9.113980958951052, 9.374187292986391, 10.24066702817719, 10.59263957106223, 11.11709670620690, 11.63355173950998, 12.07472850506848, 12.75790421968633, 13.17963442506065, 13.61327244201949, 13.98741792795436
