| L(s) = 1 | + 2-s + 4-s − 3·7-s + 8-s − 11-s + 6·13-s − 3·14-s + 16-s − 7·17-s + 5·19-s − 22-s − 6·23-s + 6·26-s − 3·28-s − 5·29-s − 3·31-s + 32-s − 7·34-s − 3·37-s + 5·38-s − 2·41-s − 4·43-s − 44-s − 6·46-s − 2·47-s + 2·49-s + 6·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.13·7-s + 0.353·8-s − 0.301·11-s + 1.66·13-s − 0.801·14-s + 1/4·16-s − 1.69·17-s + 1.14·19-s − 0.213·22-s − 1.25·23-s + 1.17·26-s − 0.566·28-s − 0.928·29-s − 0.538·31-s + 0.176·32-s − 1.20·34-s − 0.493·37-s + 0.811·38-s − 0.312·41-s − 0.609·43-s − 0.150·44-s − 0.884·46-s − 0.291·47-s + 2/7·49-s + 0.832·52-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 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 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
−7.77228867717483101427824481903, −6.94579195674683666054891651380, −6.33229646662933253912672920890, −5.82653846063764759203169478886, −4.98330350048548957559991951110, −3.86895171154843111710179552061, −3.58105947694694428667587036867, −2.57216322175076797955132720885, −1.56152608467875197108038298915, 0,
1.56152608467875197108038298915, 2.57216322175076797955132720885, 3.58105947694694428667587036867, 3.86895171154843111710179552061, 4.98330350048548957559991951110, 5.82653846063764759203169478886, 6.33229646662933253912672920890, 6.94579195674683666054891651380, 7.77228867717483101427824481903
