| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s − 4·11-s + 13-s + 4·14-s + 16-s + 17-s − 4·19-s − 20-s − 4·22-s − 4·23-s − 4·25-s + 26-s + 4·28-s + 29-s + 4·31-s + 32-s + 34-s − 4·35-s + 11·37-s − 4·38-s − 40-s − 8·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.223·20-s − 0.852·22-s − 0.834·23-s − 4/5·25-s + 0.196·26-s + 0.755·28-s + 0.185·29-s + 0.718·31-s + 0.176·32-s + 0.171·34-s − 0.676·35-s + 1.80·37-s − 0.648·38-s − 0.158·40-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.003837222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.003837222\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 41 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.89756369575031, −12.23350837978143, −11.81730489608081, −11.45177022560935, −11.13570997862890, −10.51179026498270, −10.21818400703291, −9.738914434308946, −8.847160278754480, −8.365892124851568, −8.063503580521734, −7.688610194428037, −7.290334823262571, −6.518200093388258, −6.028155011701996, −5.566418437991734, −5.007848285593412, −4.666631590603799, −4.038998581583455, −3.824857521630269, −2.876137267574712, −2.390087593967530, −1.967260762724985, −1.238322204234110, −0.4751309843442228,
0.4751309843442228, 1.238322204234110, 1.967260762724985, 2.390087593967530, 2.876137267574712, 3.824857521630269, 4.038998581583455, 4.666631590603799, 5.007848285593412, 5.566418437991734, 6.028155011701996, 6.518200093388258, 7.290334823262571, 7.688610194428037, 8.063503580521734, 8.365892124851568, 8.847160278754480, 9.738914434308946, 10.21818400703291, 10.51179026498270, 11.13570997862890, 11.45177022560935, 11.81730489608081, 12.23350837978143, 12.89756369575031
