| L(s) = 1 | + 2·7-s + 4·11-s − 6·13-s − 4·17-s − 2·19-s − 23-s − 5·25-s − 2·29-s − 4·31-s + 2·37-s − 2·41-s − 10·43-s − 3·49-s + 12·53-s − 12·59-s − 6·61-s + 10·67-s + 8·71-s − 14·73-s + 8·77-s − 10·79-s + 12·83-s + 16·89-s − 12·91-s − 10·97-s − 14·101-s + 6·103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.20·11-s − 1.66·13-s − 0.970·17-s − 0.458·19-s − 0.208·23-s − 25-s − 0.371·29-s − 0.718·31-s + 0.328·37-s − 0.312·41-s − 1.52·43-s − 3/7·49-s + 1.64·53-s − 1.56·59-s − 0.768·61-s + 1.22·67-s + 0.949·71-s − 1.63·73-s + 0.911·77-s − 1.12·79-s + 1.31·83-s + 1.69·89-s − 1.25·91-s − 1.01·97-s − 1.39·101-s + 0.591·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 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
−8.255257061974971569532486083981, −7.48370475774187535793972184411, −6.84635513230349852581658955202, −6.05124881224064209321722104454, −5.05019073254217766117424916484, −4.45542935224741786740302242155, −3.63693214406988098225814582314, −2.33698123130331617749673543091, −1.65243434960084574328445167436, 0,
1.65243434960084574328445167436, 2.33698123130331617749673543091, 3.63693214406988098225814582314, 4.45542935224741786740302242155, 5.05019073254217766117424916484, 6.05124881224064209321722104454, 6.84635513230349852581658955202, 7.48370475774187535793972184411, 8.255257061974971569532486083981
