| L(s) = 1 | + 2-s + 4-s + 5-s + 2·7-s + 8-s + 10-s − 3·11-s + 3·13-s + 2·14-s + 16-s − 17-s + 7·19-s + 20-s − 3·22-s − 4·25-s + 3·26-s + 2·28-s − 4·29-s + 31-s + 32-s − 34-s + 2·35-s − 10·37-s + 7·38-s + 40-s + 6·41-s + 6·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s + 0.353·8-s + 0.316·10-s − 0.904·11-s + 0.832·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 1.60·19-s + 0.223·20-s − 0.639·22-s − 4/5·25-s + 0.588·26-s + 0.377·28-s − 0.742·29-s + 0.179·31-s + 0.176·32-s − 0.171·34-s + 0.338·35-s − 1.64·37-s + 1.13·38-s + 0.158·40-s + 0.937·41-s + 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.528123386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.528123386\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 17 T + p T^{2} \) | 1.83.r |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−10.93419074885029781059475634406, −10.07194658168638091834826689578, −9.003607759174706982308896490394, −7.922585877134847101173520832879, −7.19025111083617417452286089024, −5.82982607301142649475740414520, −5.33284774940468326614545835445, −4.16084091864439744479801572486, −2.92707199379996070162725493955, −1.60884224963952238823179483589,
1.60884224963952238823179483589, 2.92707199379996070162725493955, 4.16084091864439744479801572486, 5.33284774940468326614545835445, 5.82982607301142649475740414520, 7.19025111083617417452286089024, 7.922585877134847101173520832879, 9.003607759174706982308896490394, 10.07194658168638091834826689578, 10.93419074885029781059475634406
