| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s + 11-s − 4·13-s − 15-s − 4·17-s − 8·19-s + 2·21-s − 4·23-s + 25-s + 27-s + 2·29-s − 8·31-s + 33-s − 2·35-s − 2·37-s − 4·39-s − 2·41-s + 6·43-s − 45-s − 12·47-s − 3·49-s − 4·51-s + 6·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.258·15-s − 0.970·17-s − 1.83·19-s + 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.174·33-s − 0.338·35-s − 0.328·37-s − 0.640·39-s − 0.312·41-s + 0.914·43-s − 0.149·45-s − 1.75·47-s − 3/7·49-s − 0.560·51-s + 0.824·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−8.515961252351078582063204366934, −7.81500633151318031543657940698, −7.07064995384853568376298678938, −6.34981381293761310343922507330, −5.14535053117062769285070670097, −4.40095938405962012615022455697, −3.78314109950025435049517073259, −2.46330191638784768361222556222, −1.81796847367624238360754427436, 0,
1.81796847367624238360754427436, 2.46330191638784768361222556222, 3.78314109950025435049517073259, 4.40095938405962012615022455697, 5.14535053117062769285070670097, 6.34981381293761310343922507330, 7.07064995384853568376298678938, 7.81500633151318031543657940698, 8.515961252351078582063204366934
