| L(s) = 1 | + 2·7-s + 11-s − 2·13-s − 6·17-s − 4·19-s + 2·29-s − 8·31-s + 2·41-s − 2·43-s − 8·47-s − 3·49-s − 8·53-s − 8·59-s − 10·61-s − 8·67-s + 12·71-s + 14·73-s + 2·77-s + 8·79-s − 6·83-s + 14·89-s − 4·91-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.301·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.371·29-s − 1.43·31-s + 0.312·41-s − 0.304·43-s − 1.16·47-s − 3/7·49-s − 1.09·53-s − 1.04·59-s − 1.28·61-s − 0.977·67-s + 1.42·71-s + 1.63·73-s + 0.227·77-s + 0.900·79-s − 0.658·83-s + 1.48·89-s − 0.419·91-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9938315506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9938315506\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 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.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−13.23743330133527, −12.68825714983071, −12.52862015730018, −11.75582146026128, −11.34516572736665, −10.92241747054963, −10.61610262770916, −9.936905058052219, −9.312742132023839, −9.014304891191558, −8.510558626391773, −7.882847088518376, −7.590237794500262, −6.891138708327164, −6.354755007115961, −6.115132449891573, −5.139040553168899, −4.774618396568024, −4.471424778673393, −3.682914433307592, −3.199041388011037, −2.221847238833802, −2.029369706391614, −1.321324048005959, −0.2811878392886174,
0.2811878392886174, 1.321324048005959, 2.029369706391614, 2.221847238833802, 3.199041388011037, 3.682914433307592, 4.471424778673393, 4.774618396568024, 5.139040553168899, 6.115132449891573, 6.354755007115961, 6.891138708327164, 7.590237794500262, 7.882847088518376, 8.510558626391773, 9.014304891191558, 9.312742132023839, 9.936905058052219, 10.61610262770916, 10.92241747054963, 11.34516572736665, 11.75582146026128, 12.52862015730018, 12.68825714983071, 13.23743330133527
