| L(s) = 1 | − 3-s − 5-s + 9-s − 6·11-s + 15-s + 4·17-s + 8·23-s + 25-s − 27-s + 2·29-s + 2·31-s + 6·33-s − 8·37-s + 2·41-s + 4·43-s − 45-s − 4·51-s + 2·53-s + 6·55-s + 10·59-s + 2·61-s − 12·67-s − 8·69-s − 14·73-s − 75-s − 2·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.80·11-s + 0.258·15-s + 0.970·17-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.359·31-s + 1.04·33-s − 1.31·37-s + 0.312·41-s + 0.609·43-s − 0.149·45-s − 0.560·51-s + 0.274·53-s + 0.809·55-s + 1.30·59-s + 0.256·61-s − 1.46·67-s − 0.963·69-s − 1.63·73-s − 0.115·75-s − 0.225·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−14.01604755436169, −13.40819038671527, −13.01197714467251, −12.66513383894273, −11.99817661265313, −11.74559610370769, −10.96932064506553, −10.65604680674988, −10.27004792556075, −9.775089808466737, −9.011222662781599, −8.531104401283314, −7.982609800091472, −7.408451228488919, −7.156834064063197, −6.464269703974976, −5.666723951353769, −5.371833062529921, −4.884821463459689, −4.324736501392006, −3.516331929095292, −2.931524367959786, −2.493057667601195, −1.462243464291978, −0.7699993034151816, 0,
0.7699993034151816, 1.462243464291978, 2.493057667601195, 2.931524367959786, 3.516331929095292, 4.324736501392006, 4.884821463459689, 5.371833062529921, 5.666723951353769, 6.464269703974976, 7.156834064063197, 7.408451228488919, 7.982609800091472, 8.531104401283314, 9.011222662781599, 9.775089808466737, 10.27004792556075, 10.65604680674988, 10.96932064506553, 11.74559610370769, 11.99817661265313, 12.66513383894273, 13.01197714467251, 13.40819038671527, 14.01604755436169
