| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s − 4·11-s + 13-s + 16-s + 2·17-s − 19-s + 2·20-s − 4·22-s + 8·23-s − 25-s + 26-s − 2·29-s + 32-s + 2·34-s + 10·37-s − 38-s + 2·40-s − 2·41-s + 12·43-s − 4·44-s + 8·46-s + 4·47-s − 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s − 1.20·11-s + 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.229·19-s + 0.447·20-s − 0.852·22-s + 1.66·23-s − 1/5·25-s + 0.196·26-s − 0.371·29-s + 0.176·32-s + 0.342·34-s + 1.64·37-s − 0.162·38-s + 0.316·40-s − 0.312·41-s + 1.82·43-s − 0.603·44-s + 1.17·46-s + 0.583·47-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 217854 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 217854 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 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.ac |
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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
−13.22146240818266, −12.81287028099180, −12.54435300200169, −11.86781929396338, −11.31330155730303, −10.88062647683002, −10.48191389432741, −10.09928805963842, −9.461877219225148, −9.035837582319506, −8.586705924530868, −7.744754826872026, −7.490342685265160, −7.096401781270909, −6.184096142614225, −5.865684968562340, −5.691395236687215, −4.881213354001648, −4.581542731969378, −3.962196248731661, −3.103121231463941, −2.803120199672129, −2.341064803967873, −1.529934167242605, −1.039744888893497, 0,
1.039744888893497, 1.529934167242605, 2.341064803967873, 2.803120199672129, 3.103121231463941, 3.962196248731661, 4.581542731969378, 4.881213354001648, 5.691395236687215, 5.865684968562340, 6.184096142614225, 7.096401781270909, 7.490342685265160, 7.744754826872026, 8.586705924530868, 9.035837582319506, 9.461877219225148, 10.09928805963842, 10.48191389432741, 10.88062647683002, 11.31330155730303, 11.86781929396338, 12.54435300200169, 12.81287028099180, 13.22146240818266
