| L(s) = 1 | − 7-s + 11-s − 4·13-s − 3·17-s + 5·19-s − 3·23-s + 6·29-s + 8·31-s − 7·37-s − 9·41-s + 8·43-s + 3·47-s − 6·49-s − 6·53-s − 3·59-s + 14·61-s + 2·67-s + 9·71-s + 2·73-s − 77-s − 79-s − 12·83-s − 18·89-s + 4·91-s + 11·97-s − 15·101-s − 4·103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.301·11-s − 1.10·13-s − 0.727·17-s + 1.14·19-s − 0.625·23-s + 1.11·29-s + 1.43·31-s − 1.15·37-s − 1.40·41-s + 1.21·43-s + 0.437·47-s − 6/7·49-s − 0.824·53-s − 0.390·59-s + 1.79·61-s + 0.244·67-s + 1.06·71-s + 0.234·73-s − 0.113·77-s − 0.112·79-s − 1.31·83-s − 1.90·89-s + 0.419·91-s + 1.11·97-s − 1.49·101-s − 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−7.14331538556334380263942154793, −6.77551349038803816501864820579, −6.03300552574440343319107329413, −5.16044905513780578872463830693, −4.64778672607481512485460030243, −3.78821065015516854077433961974, −2.95303903856930701750739446222, −2.28540792900314169787973438450, −1.17356821519877114103262626479, 0,
1.17356821519877114103262626479, 2.28540792900314169787973438450, 2.95303903856930701750739446222, 3.78821065015516854077433961974, 4.64778672607481512485460030243, 5.16044905513780578872463830693, 6.03300552574440343319107329413, 6.77551349038803816501864820579, 7.14331538556334380263942154793
