| L(s) = 1 | + 4·11-s + 13-s − 6·17-s + 8·19-s + 8·23-s + 2·29-s + 8·31-s − 10·37-s − 6·41-s − 4·43-s − 7·49-s + 14·53-s − 12·59-s + 10·61-s − 8·67-s + 14·73-s − 4·79-s − 4·83-s + 10·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 0.277·13-s − 1.45·17-s + 1.83·19-s + 1.66·23-s + 0.371·29-s + 1.43·31-s − 1.64·37-s − 0.937·41-s − 0.609·43-s − 49-s + 1.92·53-s − 1.56·59-s + 1.28·61-s − 0.977·67-s + 1.63·73-s − 0.450·79-s − 0.439·83-s + 1.05·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.50896907168993, −13.03642958696245, −12.17579587338324, −11.99835599570777, −11.49295317508203, −11.14990735926587, −10.50793961640934, −10.09426237993159, −9.416081559547135, −9.151849458143568, −8.620782712959287, −8.283881955170742, −7.504853046304993, −6.852982829180321, −6.786143764988213, −6.244025297335607, −5.416482709193815, −5.012994203118976, −4.573503252804360, −3.845134564635994, −3.368420192228452, −2.871999219804087, −2.150108401295796, −1.287310615407548, −1.058193817842103, 0,
1.058193817842103, 1.287310615407548, 2.150108401295796, 2.871999219804087, 3.368420192228452, 3.845134564635994, 4.573503252804360, 5.012994203118976, 5.416482709193815, 6.244025297335607, 6.786143764988213, 6.852982829180321, 7.504853046304993, 8.283881955170742, 8.620782712959287, 9.151849458143568, 9.416081559547135, 10.09426237993159, 10.50793961640934, 11.14990735926587, 11.49295317508203, 11.99835599570777, 12.17579587338324, 13.03642958696245, 13.50896907168993
