| L(s) = 1 | − 3·9-s + 11-s + 2·13-s + 6·17-s − 4·19-s + 8·23-s − 6·29-s − 2·37-s − 6·41-s + 8·43-s + 8·47-s − 7·49-s + 6·53-s + 8·59-s + 10·61-s + 4·67-s − 8·71-s − 10·73-s + 79-s + 9·81-s − 4·83-s + 10·89-s − 2·97-s − 3·99-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 9-s + 0.301·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 1.66·23-s − 1.11·29-s − 0.328·37-s − 0.937·41-s + 1.21·43-s + 1.16·47-s − 49-s + 0.824·53-s + 1.04·59-s + 1.28·61-s + 0.488·67-s − 0.949·71-s − 1.17·73-s + 0.112·79-s + 81-s − 0.439·83-s + 1.05·89-s − 0.203·97-s − 0.301·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 347600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 347600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−12.80755205102147, −12.27307181231439, −11.86967785341780, −11.38868849695563, −10.99674904762431, −10.61942309432593, −10.08610347644735, −9.578549630425173, −8.900535354929850, −8.857398290763957, −8.230760069049784, −7.818162567866192, −7.076966923008847, −6.946783179125705, −6.128237103147388, −5.765748529043784, −5.368780782423278, −4.899480100898060, −4.121568017435818, −3.669901797075668, −3.227914749964772, −2.651541039797436, −2.099397973033889, −1.294020275107597, −0.8355332864568439, 0,
0.8355332864568439, 1.294020275107597, 2.099397973033889, 2.651541039797436, 3.227914749964772, 3.669901797075668, 4.121568017435818, 4.899480100898060, 5.368780782423278, 5.765748529043784, 6.128237103147388, 6.946783179125705, 7.076966923008847, 7.818162567866192, 8.230760069049784, 8.857398290763957, 8.900535354929850, 9.578549630425173, 10.08610347644735, 10.61942309432593, 10.99674904762431, 11.38868849695563, 11.86967785341780, 12.27307181231439, 12.80755205102147
