| L(s) = 1 | − 3-s − 7-s + 9-s − 6·13-s − 17-s + 6·19-s + 21-s + 2·23-s − 27-s + 2·29-s − 8·31-s − 4·37-s + 6·39-s + 10·41-s − 6·43-s + 8·47-s + 49-s + 51-s + 12·53-s − 6·57-s + 4·59-s − 4·61-s − 63-s − 10·67-s − 2·69-s − 6·73-s − 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.242·17-s + 1.37·19-s + 0.218·21-s + 0.417·23-s − 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.657·37-s + 0.960·39-s + 1.56·41-s − 0.914·43-s + 1.16·47-s + 1/7·49-s + 0.140·51-s + 1.64·53-s − 0.794·57-s + 0.520·59-s − 0.512·61-s − 0.125·63-s − 1.22·67-s − 0.240·69-s − 0.702·73-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.063596266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.063596266\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.92945975901190, −14.46564554344203, −13.85882995350342, −13.29376615266771, −12.76053893201138, −12.15792894742196, −11.92976638470658, −11.29768085694515, −10.59519540322880, −10.23615765253381, −9.526194452636085, −9.254759496418454, −8.552944210385648, −7.598804488963832, −7.262874508197664, −6.939325497831495, −6.011247452385401, −5.488821648445463, −5.037159335043510, −4.371220491421654, −3.654925556211697, −2.865671583644946, −2.291113537762914, −1.317094239770093, −0.4142488174418841,
0.4142488174418841, 1.317094239770093, 2.291113537762914, 2.865671583644946, 3.654925556211697, 4.371220491421654, 5.037159335043510, 5.488821648445463, 6.011247452385401, 6.939325497831495, 7.262874508197664, 7.598804488963832, 8.552944210385648, 9.254759496418454, 9.526194452636085, 10.23615765253381, 10.59519540322880, 11.29768085694515, 11.92976638470658, 12.15792894742196, 12.76053893201138, 13.29376615266771, 13.85882995350342, 14.46564554344203, 14.92945975901190
