| L(s) = 1 | − 3-s + 2·7-s − 2·9-s + 13-s − 6·17-s − 2·19-s − 2·21-s + 5·27-s − 3·29-s + 5·31-s + 8·37-s − 39-s + 3·41-s + 8·43-s − 9·47-s − 3·49-s + 6·51-s + 6·53-s + 2·57-s − 12·59-s − 14·61-s − 4·63-s + 8·67-s − 15·71-s + 7·73-s + 10·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s − 2/3·9-s + 0.277·13-s − 1.45·17-s − 0.458·19-s − 0.436·21-s + 0.962·27-s − 0.557·29-s + 0.898·31-s + 1.31·37-s − 0.160·39-s + 0.468·41-s + 1.21·43-s − 1.31·47-s − 3/7·49-s + 0.840·51-s + 0.824·53-s + 0.264·57-s − 1.56·59-s − 1.79·61-s − 0.503·63-s + 0.977·67-s − 1.78·71-s + 0.819·73-s + 1.12·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.69122708523959, −14.25907094669198, −13.60843168722334, −13.29684948973470, −12.55042203109649, −12.16366962632609, −11.41938717826240, −11.14129019403651, −10.88917886459460, −10.21321014224018, −9.436531358539509, −8.973269881507822, −8.473564313089464, −7.904249036862902, −7.434295934055289, −6.527776529132674, −6.225486149526487, −5.745690253711411, −4.831941986413777, −4.648719140424236, −3.962529254602463, −3.046762145314769, −2.430018610790448, −1.751356717441326, −0.8520042887497565, 0,
0.8520042887497565, 1.751356717441326, 2.430018610790448, 3.046762145314769, 3.962529254602463, 4.648719140424236, 4.831941986413777, 5.745690253711411, 6.225486149526487, 6.527776529132674, 7.434295934055289, 7.904249036862902, 8.473564313089464, 8.973269881507822, 9.436531358539509, 10.21321014224018, 10.88917886459460, 11.14129019403651, 11.41938717826240, 12.16366962632609, 12.55042203109649, 13.29684948973470, 13.60843168722334, 14.25907094669198, 14.69122708523959
