| L(s) = 1 | + 3-s − 2·4-s − 5-s − 7-s + 9-s + 5·11-s − 2·12-s − 5·13-s − 15-s + 4·16-s − 5·19-s + 2·20-s − 21-s + 23-s − 4·25-s + 27-s + 2·28-s + 6·29-s + 6·31-s + 5·33-s + 35-s − 2·36-s − 4·37-s − 5·39-s − 7·41-s − 7·43-s − 10·44-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.577·12-s − 1.38·13-s − 0.258·15-s + 16-s − 1.14·19-s + 0.447·20-s − 0.218·21-s + 0.208·23-s − 4/5·25-s + 0.192·27-s + 0.377·28-s + 1.11·29-s + 1.07·31-s + 0.870·33-s + 0.169·35-s − 1/3·36-s − 0.657·37-s − 0.800·39-s − 1.09·41-s − 1.06·43-s − 1.50·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.349285973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.349285973\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−8.400305546400304934198540436606, −7.35343306002931313684207034984, −6.80157352545949033063510606567, −5.99836690419139153065721702310, −4.91133038693438092528484452365, −4.34359280024281907660931435832, −3.76877302669386175662975373167, −2.94513597171598774414644542529, −1.84939129269636245262064680737, −0.59424854754834913087020398484,
0.59424854754834913087020398484, 1.84939129269636245262064680737, 2.94513597171598774414644542529, 3.76877302669386175662975373167, 4.34359280024281907660931435832, 4.91133038693438092528484452365, 5.99836690419139153065721702310, 6.80157352545949033063510606567, 7.35343306002931313684207034984, 8.400305546400304934198540436606
