| L(s) = 1 | − 2-s + 4-s + 2·5-s + 4·7-s − 8-s − 2·10-s + 2·13-s − 4·14-s + 16-s − 2·17-s + 2·20-s − 4·23-s − 25-s − 2·26-s + 4·28-s + 29-s + 6·31-s − 32-s + 2·34-s + 8·35-s − 4·37-s − 2·40-s − 2·41-s + 4·43-s + 4·46-s + 8·47-s + 9·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s + 1.51·7-s − 0.353·8-s − 0.632·10-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.447·20-s − 0.834·23-s − 1/5·25-s − 0.392·26-s + 0.755·28-s + 0.185·29-s + 1.07·31-s − 0.176·32-s + 0.342·34-s + 1.35·35-s − 0.657·37-s − 0.316·40-s − 0.312·41-s + 0.609·43-s + 0.589·46-s + 1.16·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.420369758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.420369758\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 29 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−10.72322127064309807787803868303, −10.05678878752383322495247879470, −8.981711256048709126985302324480, −8.332760238166233719069742869306, −7.45444572661170361596851801094, −6.27877814871073397422537784950, −5.41372438913065560036006513032, −4.21883405823766709165029813519, −2.39785286669760081428744405618, −1.41073224367500160090044037897,
1.41073224367500160090044037897, 2.39785286669760081428744405618, 4.21883405823766709165029813519, 5.41372438913065560036006513032, 6.27877814871073397422537784950, 7.45444572661170361596851801094, 8.332760238166233719069742869306, 8.981711256048709126985302324480, 10.05678878752383322495247879470, 10.72322127064309807787803868303
