| L(s) = 1 | + 2-s − 2·3-s + 4-s − 3·5-s − 2·6-s − 4·7-s + 8-s + 9-s − 3·10-s − 2·11-s − 2·12-s + 4·13-s − 4·14-s + 6·15-s + 16-s − 2·17-s + 18-s − 2·19-s − 3·20-s + 8·21-s − 2·22-s + 23-s − 2·24-s + 4·25-s + 4·26-s + 4·27-s − 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.34·5-s − 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.603·11-s − 0.577·12-s + 1.10·13-s − 1.06·14-s + 1.54·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.458·19-s − 0.670·20-s + 1.74·21-s − 0.426·22-s + 0.208·23-s − 0.408·24-s + 4/5·25-s + 0.784·26-s + 0.769·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 107 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−11.79234987333156392289715011404, −11.18552664455231013682836269463, −10.37720443838371107887581089839, −8.833954583443673316861058339108, −7.44658367022760519929025644109, −6.45990321941711893864042534840, −5.64485830848512435812878169299, −4.23072385247416176825636829094, −3.23388376939178278745840144420, 0,
3.23388376939178278745840144420, 4.23072385247416176825636829094, 5.64485830848512435812878169299, 6.45990321941711893864042534840, 7.44658367022760519929025644109, 8.833954583443673316861058339108, 10.37720443838371107887581089839, 11.18552664455231013682836269463, 11.79234987333156392289715011404
