| L(s) = 1 | + 2·7-s + 2·13-s + 17-s + 4·19-s − 2·23-s − 5·25-s + 6·31-s + 10·41-s + 4·43-s + 4·47-s − 3·49-s + 2·53-s + 4·59-s + 4·67-s + 2·71-s − 14·73-s + 6·79-s + 12·83-s + 2·89-s + 4·91-s − 2·97-s + 2·101-s + 4·103-s + 16·107-s − 4·109-s + 6·113-s + 2·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.554·13-s + 0.242·17-s + 0.917·19-s − 0.417·23-s − 25-s + 1.07·31-s + 1.56·41-s + 0.609·43-s + 0.583·47-s − 3/7·49-s + 0.274·53-s + 0.520·59-s + 0.488·67-s + 0.237·71-s − 1.63·73-s + 0.675·79-s + 1.31·83-s + 0.211·89-s + 0.419·91-s − 0.203·97-s + 0.199·101-s + 0.394·103-s + 1.54·107-s − 0.383·109-s + 0.564·113-s + 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.877616707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.877616707\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−9.727333497988217362385853707969, −8.895695976213285256038805237837, −7.985985062756261177697988597093, −7.48540845964688300784548369243, −6.28428132320064706038813400594, −5.54303189496107426211899879195, −4.55493862607580750994390745098, −3.63500318653539816364252970235, −2.37692569999125749532190868382, −1.10022247683164928183821192985,
1.10022247683164928183821192985, 2.37692569999125749532190868382, 3.63500318653539816364252970235, 4.55493862607580750994390745098, 5.54303189496107426211899879195, 6.28428132320064706038813400594, 7.48540845964688300784548369243, 7.985985062756261177697988597093, 8.895695976213285256038805237837, 9.727333497988217362385853707969
