| 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
| \(L(1)\) |
\(\approx\) |
\(2.253591182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.253591182\) |
| \(L(\frac{3}{2})\) |
|
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.c |
| 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.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 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.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 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.q |
| 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.c |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−11.23088681177259931213600888425, −10.31022962994137309665668010111, −8.850929506389471866653432728217, −7.978909275244120811387358467011, −7.40719357170324726064049984804, −6.12027842085336087630745331402, −4.98617740641702215386206821415, −4.29411921029809728013580242122, −3.13738525881404839811628728846, −1.49549994584612613248947406224,
1.49549994584612613248947406224, 3.13738525881404839811628728846, 4.29411921029809728013580242122, 4.98617740641702215386206821415, 6.12027842085336087630745331402, 7.40719357170324726064049984804, 7.978909275244120811387358467011, 8.850929506389471866653432728217, 10.31022962994137309665668010111, 11.23088681177259931213600888425
