| L(s) = 1 | − 4·5-s − 7-s − 3·9-s − 4·11-s − 4·13-s + 2·19-s + 11·25-s − 6·29-s + 2·31-s + 4·35-s − 37-s + 2·41-s + 8·43-s + 12·45-s + 4·47-s + 49-s − 6·53-s + 16·55-s + 6·59-s + 3·63-s + 16·65-s − 4·67-s − 8·71-s + 10·73-s + 4·77-s − 8·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.377·7-s − 9-s − 1.20·11-s − 1.10·13-s + 0.458·19-s + 11/5·25-s − 1.11·29-s + 0.359·31-s + 0.676·35-s − 0.164·37-s + 0.312·41-s + 1.21·43-s + 1.78·45-s + 0.583·47-s + 1/7·49-s − 0.824·53-s + 2.15·55-s + 0.781·59-s + 0.377·63-s + 1.98·65-s − 0.488·67-s − 0.949·71-s + 1.17·73-s + 0.455·77-s − 0.900·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4188355943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4188355943\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 37 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−8.962038555811940208667092331639, −8.198061734199791289115955710221, −7.57167178089628656871075734507, −7.13788706896658819557844494948, −5.83193960073228849985985080649, −5.05203485694549250692665439486, −4.18437962166128173929044575331, −3.21090636476466589268085715680, −2.56476155909250453696337944857, −0.39765703754171760008269461983,
0.39765703754171760008269461983, 2.56476155909250453696337944857, 3.21090636476466589268085715680, 4.18437962166128173929044575331, 5.05203485694549250692665439486, 5.83193960073228849985985080649, 7.13788706896658819557844494948, 7.57167178089628656871075734507, 8.198061734199791289115955710221, 8.962038555811940208667092331639
