| L(s) = 1 | + 2·7-s − 5·11-s + 4·13-s − 6·17-s − 7·19-s − 4·23-s + 5·29-s + 3·31-s − 2·37-s + 7·41-s − 6·43-s + 6·47-s − 3·49-s + 10·53-s − 15·59-s − 14·61-s + 4·67-s + 5·71-s − 14·73-s − 10·77-s + 8·79-s + 14·83-s + 3·89-s + 8·91-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.50·11-s + 1.10·13-s − 1.45·17-s − 1.60·19-s − 0.834·23-s + 0.928·29-s + 0.538·31-s − 0.328·37-s + 1.09·41-s − 0.914·43-s + 0.875·47-s − 3/7·49-s + 1.37·53-s − 1.95·59-s − 1.79·61-s + 0.488·67-s + 0.593·71-s − 1.63·73-s − 1.13·77-s + 0.900·79-s + 1.53·83-s + 0.317·89-s + 0.838·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 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
−13.74244449503144, −13.20263552450193, −12.94766962525565, −12.24229730870824, −11.86607430725351, −11.08786207178903, −10.84655002473034, −10.50387106125530, −10.07247585016579, −9.187247729924644, −8.662313198585261, −8.470377634647546, −7.850209919893827, −7.513945014225045, −6.672434129333824, −6.173066757229306, −5.924225316257671, −5.016498524343317, −4.590459198245138, −4.275756555028535, −3.479309431914351, −2.759636549324124, −2.143581593499180, −1.799478464373610, −0.7605841431519723, 0,
0.7605841431519723, 1.799478464373610, 2.143581593499180, 2.759636549324124, 3.479309431914351, 4.275756555028535, 4.590459198245138, 5.016498524343317, 5.924225316257671, 6.173066757229306, 6.672434129333824, 7.513945014225045, 7.850209919893827, 8.470377634647546, 8.662313198585261, 9.187247729924644, 10.07247585016579, 10.50387106125530, 10.84655002473034, 11.08786207178903, 11.86607430725351, 12.24229730870824, 12.94766962525565, 13.20263552450193, 13.74244449503144
