| L(s) = 1 | + 3-s + 7-s + 9-s − 2·11-s − 2·13-s − 17-s + 5·19-s + 21-s − 4·23-s + 27-s + 9·29-s − 2·31-s − 2·33-s − 6·37-s − 2·39-s + 8·41-s − 12·43-s − 8·47-s + 49-s − 51-s − 10·53-s + 5·57-s + 15·61-s + 63-s − 10·67-s − 4·69-s − 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.242·17-s + 1.14·19-s + 0.218·21-s − 0.834·23-s + 0.192·27-s + 1.67·29-s − 0.359·31-s − 0.348·33-s − 0.986·37-s − 0.320·39-s + 1.24·41-s − 1.82·43-s − 1.16·47-s + 1/7·49-s − 0.140·51-s − 1.37·53-s + 0.662·57-s + 1.92·61-s + 0.125·63-s − 1.22·67-s − 0.481·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.530695387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.530695387\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 15 T + p T^{2} \) | 1.61.ap |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.79747923846836, −14.24766837552465, −14.12680634960552, −13.29892935726395, −13.00281762321112, −12.24673476336066, −11.81301962329101, −11.32065789709487, −10.56366023243684, −9.949711701716565, −9.824749469751435, −8.892950791652243, −8.481438559000264, −7.886900767915667, −7.464570680374822, −6.839291004659055, −6.181344734926901, −5.416272822292424, −4.824053518314557, −4.419986408610175, −3.380411611902854, −3.052359119853336, −2.180838902185656, −1.606470191481376, −0.5612559084387898,
0.5612559084387898, 1.606470191481376, 2.180838902185656, 3.052359119853336, 3.380411611902854, 4.419986408610175, 4.824053518314557, 5.416272822292424, 6.181344734926901, 6.839291004659055, 7.464570680374822, 7.886900767915667, 8.481438559000264, 8.892950791652243, 9.824749469751435, 9.949711701716565, 10.56366023243684, 11.32065789709487, 11.81301962329101, 12.24673476336066, 13.00281762321112, 13.29892935726395, 14.12680634960552, 14.24766837552465, 14.79747923846836
