| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 11-s − 2·13-s + 2·14-s + 16-s + 6·17-s + 2·19-s + 22-s + 2·26-s − 2·28-s − 6·29-s − 4·31-s − 32-s − 6·34-s − 2·37-s − 2·38-s + 6·41-s + 10·43-s − 44-s − 12·47-s − 3·49-s − 2·52-s + 12·53-s + 2·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.301·11-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.458·19-s + 0.213·22-s + 0.392·26-s − 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.324·38-s + 0.937·41-s + 1.52·43-s − 0.150·44-s − 1.75·47-s − 3/7·49-s − 0.277·52-s + 1.64·53-s + 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−7.66442610206164190077502590390, −7.48965659088412414475096394949, −6.57203223801687614334384416013, −5.73139720572849869792987630684, −5.21289672268637490430270652412, −3.93138628346159363239287072337, −3.19350365088848575516553636134, −2.36876360541926021980240488969, −1.20545664759296527865377311925, 0,
1.20545664759296527865377311925, 2.36876360541926021980240488969, 3.19350365088848575516553636134, 3.93138628346159363239287072337, 5.21289672268637490430270652412, 5.73139720572849869792987630684, 6.57203223801687614334384416013, 7.48965659088412414475096394949, 7.66442610206164190077502590390
