| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s − 3·11-s + 2·13-s + 2·14-s + 16-s − 3·17-s + 19-s − 3·22-s + 6·23-s − 5·25-s + 2·26-s + 2·28-s + 4·31-s + 32-s − 3·34-s + 4·37-s + 38-s + 9·41-s + 43-s − 3·44-s + 6·46-s − 6·47-s − 3·49-s − 5·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 0.904·11-s + 0.554·13-s + 0.534·14-s + 1/4·16-s − 0.727·17-s + 0.229·19-s − 0.639·22-s + 1.25·23-s − 25-s + 0.392·26-s + 0.377·28-s + 0.718·31-s + 0.176·32-s − 0.514·34-s + 0.657·37-s + 0.162·38-s + 1.40·41-s + 0.152·43-s − 0.452·44-s + 0.884·46-s − 0.875·47-s − 3/7·49-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−13.68568684885488, −13.07447401038030, −12.93169226250630, −12.36788060672457, −11.58856054896557, −11.37465800210448, −10.89243831409937, −10.58727409028141, −9.801916428444692, −9.379843684858276, −8.769000086430048, −8.089267674869550, −7.851425689515433, −7.343059240072015, −6.609544186884536, −6.200236418647770, −5.651216044044289, −5.034066023706969, −4.668272497562163, −4.195324552402819, −3.448889067589547, −2.865926403729747, −2.371541267080036, −1.637092217828908, −1.035109344854876, 0,
1.035109344854876, 1.637092217828908, 2.371541267080036, 2.865926403729747, 3.448889067589547, 4.195324552402819, 4.668272497562163, 5.034066023706969, 5.651216044044289, 6.200236418647770, 6.609544186884536, 7.343059240072015, 7.851425689515433, 8.089267674869550, 8.769000086430048, 9.379843684858276, 9.801916428444692, 10.58727409028141, 10.89243831409937, 11.37465800210448, 11.58856054896557, 12.36788060672457, 12.93169226250630, 13.07447401038030, 13.68568684885488
