| L(s) = 1 | + 3-s + 7-s + 9-s + 3·11-s + 13-s − 3·17-s − 7·19-s + 21-s − 3·23-s − 5·25-s + 27-s − 2·31-s + 3·33-s − 37-s + 39-s − 6·41-s − 4·43-s − 6·47-s − 6·49-s − 3·51-s − 9·53-s − 7·57-s + 10·61-s + 63-s + 2·67-s − 3·69-s − 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 0.727·17-s − 1.60·19-s + 0.218·21-s − 0.625·23-s − 25-s + 0.192·27-s − 0.359·31-s + 0.522·33-s − 0.164·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s − 0.875·47-s − 6/7·49-s − 0.420·51-s − 1.23·53-s − 0.927·57-s + 1.28·61-s + 0.125·63-s + 0.244·67-s − 0.361·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 37 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−7.77794599625144387841353120869, −6.66826090398918076260991448252, −6.49494835392927006659413256383, −5.46983042730634014347435719418, −4.48702773694669977478193656749, −4.02500517572888180323746344706, −3.22838831488081299369022792445, −2.06885843983646953069448344954, −1.61623514812598456008395519291, 0,
1.61623514812598456008395519291, 2.06885843983646953069448344954, 3.22838831488081299369022792445, 4.02500517572888180323746344706, 4.48702773694669977478193656749, 5.46983042730634014347435719418, 6.49494835392927006659413256383, 6.66826090398918076260991448252, 7.77794599625144387841353120869
