| L(s) = 1 | − 2-s + 4-s − 2·5-s − 7-s − 8-s + 2·10-s − 13-s + 14-s + 16-s + 2·17-s + 4·19-s − 2·20-s − 25-s + 26-s − 28-s + 6·29-s + 8·31-s − 32-s − 2·34-s + 2·35-s + 6·37-s − 4·38-s + 2·40-s + 2·41-s − 8·43-s + 49-s + 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s − 0.353·8-s + 0.632·10-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.447·20-s − 1/5·25-s + 0.196·26-s − 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.342·34-s + 0.338·35-s + 0.986·37-s − 0.648·38-s + 0.316·40-s + 0.312·41-s − 1.21·43-s + 1/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−13.33424370191005, −12.70984137023222, −12.11194165169849, −11.81000300133288, −11.62339088483252, −10.93534177577423, −10.35593222930268, −9.971421690357015, −9.639744940735299, −9.047316708264227, −8.430752507544920, −8.094641209864898, −7.710405499010008, −7.059593169964462, −6.840077411636874, −6.058751209191454, −5.701834272642647, −4.933969290962747, −4.415865865667605, −3.881714560921548, −3.108385428826073, −2.906401228739157, −2.146106668390494, −1.225812371770995, −0.7793198609558972, 0,
0.7793198609558972, 1.225812371770995, 2.146106668390494, 2.906401228739157, 3.108385428826073, 3.881714560921548, 4.415865865667605, 4.933969290962747, 5.701834272642647, 6.058751209191454, 6.840077411636874, 7.059593169964462, 7.710405499010008, 8.094641209864898, 8.430752507544920, 9.047316708264227, 9.639744940735299, 9.971421690357015, 10.35593222930268, 10.93534177577423, 11.62339088483252, 11.81000300133288, 12.11194165169849, 12.70984137023222, 13.33424370191005
