| L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s − 2·11-s + 13-s + 15-s + 17-s + 8·19-s + 21-s + 6·23-s − 4·25-s + 5·27-s + 6·29-s − 8·31-s + 2·33-s + 35-s − 37-s − 39-s − 4·41-s + 5·43-s + 2·45-s + 11·47-s − 6·49-s − 51-s − 6·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.603·11-s + 0.277·13-s + 0.258·15-s + 0.242·17-s + 1.83·19-s + 0.218·21-s + 1.25·23-s − 4/5·25-s + 0.962·27-s + 1.11·29-s − 1.43·31-s + 0.348·33-s + 0.169·35-s − 0.164·37-s − 0.160·39-s − 0.624·41-s + 0.762·43-s + 0.298·45-s + 1.60·47-s − 6/7·49-s − 0.140·51-s − 0.824·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.199073624914521124879144650154, −7.45006616547868092938112607192, −6.84264315503396464710701677151, −5.73008352562130164469600506696, −5.44533999780262362186459034761, −4.47795141458358907708340486916, −3.33468999549701088227176494693, −2.82245043350117247777660171896, −1.21571742908228392623596644749, 0,
1.21571742908228392623596644749, 2.82245043350117247777660171896, 3.33468999549701088227176494693, 4.47795141458358907708340486916, 5.44533999780262362186459034761, 5.73008352562130164469600506696, 6.84264315503396464710701677151, 7.45006616547868092938112607192, 8.199073624914521124879144650154
