| L(s) = 1 | + 7-s − 11-s − 3·13-s − 5·19-s + 4·23-s − 5·25-s + 4·29-s − 4·31-s − 37-s − 8·41-s − 8·47-s − 6·49-s − 4·53-s + 8·59-s + 7·61-s − 5·67-s − 11·73-s − 77-s − 79-s + 16·83-s − 3·91-s + 5·97-s − 4·101-s − 7·103-s + 8·107-s − 14·109-s − 16·113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.301·11-s − 0.832·13-s − 1.14·19-s + 0.834·23-s − 25-s + 0.742·29-s − 0.718·31-s − 0.164·37-s − 1.24·41-s − 1.16·47-s − 6/7·49-s − 0.549·53-s + 1.04·59-s + 0.896·61-s − 0.610·67-s − 1.28·73-s − 0.113·77-s − 0.112·79-s + 1.75·83-s − 0.314·91-s + 0.507·97-s − 0.398·101-s − 0.689·103-s + 0.773·107-s − 1.34·109-s − 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−8.491643667319512722844908355271, −7.911378237636725425197479413896, −7.03965186684405176147162165641, −6.34478365581971936012284535729, −5.27713728477525503809337925520, −4.71354844510917304529360302350, −3.69776450266362512034508615687, −2.61846300518308503839477032545, −1.66629105944210080601378298402, 0,
1.66629105944210080601378298402, 2.61846300518308503839477032545, 3.69776450266362512034508615687, 4.71354844510917304529360302350, 5.27713728477525503809337925520, 6.34478365581971936012284535729, 7.03965186684405176147162165641, 7.911378237636725425197479413896, 8.491643667319512722844908355271
