| L(s) = 1 | + 2·5-s + 4·13-s − 19-s − 8·23-s − 25-s + 2·29-s + 2·31-s − 8·37-s + 2·41-s + 4·43-s + 4·47-s − 7·49-s + 2·53-s + 10·61-s + 8·65-s + 16·71-s − 6·73-s + 14·79-s − 6·83-s − 18·89-s − 2·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.10·13-s − 0.229·19-s − 1.66·23-s − 1/5·25-s + 0.371·29-s + 0.359·31-s − 1.31·37-s + 0.312·41-s + 0.609·43-s + 0.583·47-s − 49-s + 0.274·53-s + 1.28·61-s + 0.992·65-s + 1.89·71-s − 0.702·73-s + 1.57·79-s − 0.658·83-s − 1.90·89-s − 0.205·95-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 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
−12.74234937334425, −12.47551738929657, −11.93833846902573, −11.41374253500018, −11.01432268244880, −10.43733676618030, −10.14064635168550, −9.657208041523554, −9.255958628268783, −8.669267407875587, −8.189774538363076, −7.992733203841016, −7.141330379325052, −6.718561376105433, −6.221653201534837, −5.812160424524036, −5.483985418089738, −4.827344288396757, −4.173037185730986, −3.774057048537376, −3.257455996449206, −2.427890238706909, −2.082305705560407, −1.495277710920840, −0.8630037570607414, 0,
0.8630037570607414, 1.495277710920840, 2.082305705560407, 2.427890238706909, 3.257455996449206, 3.774057048537376, 4.173037185730986, 4.827344288396757, 5.483985418089738, 5.812160424524036, 6.221653201534837, 6.718561376105433, 7.141330379325052, 7.992733203841016, 8.189774538363076, 8.669267407875587, 9.255958628268783, 9.657208041523554, 10.14064635168550, 10.43733676618030, 11.01432268244880, 11.41374253500018, 11.93833846902573, 12.47551738929657, 12.74234937334425
