| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 13-s + 15-s + 5·17-s − 19-s + 21-s + 25-s − 27-s − 29-s − 10·31-s + 35-s + 7·37-s − 39-s − 2·41-s + 6·43-s − 45-s + 6·47-s − 6·49-s − 5·51-s − 8·53-s + 57-s − 4·59-s − 4·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s + 1.21·17-s − 0.229·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s − 0.185·29-s − 1.79·31-s + 0.169·35-s + 1.15·37-s − 0.160·39-s − 0.312·41-s + 0.914·43-s − 0.149·45-s + 0.875·47-s − 6/7·49-s − 0.700·51-s − 1.09·53-s + 0.132·57-s − 0.520·59-s − 0.512·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 17 T + p T^{2} \) | 1.83.ar |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.93994787619157, −13.16783879904174, −12.72087861214193, −12.48701948760310, −11.94576343738245, −11.33341414772475, −10.98265783514805, −10.59174104215288, −9.916850756434929, −9.451655452775486, −9.096052512656010, −8.301848154150133, −7.838991221373860, −7.389627717874232, −6.909643929291096, −6.105543104885243, −5.971890284878268, −5.197004172516070, −4.800952752598403, −3.955596757145412, −3.651836888154007, −3.020443185172863, −2.238964273675949, −1.457028096481633, −0.7794267336889367, 0,
0.7794267336889367, 1.457028096481633, 2.238964273675949, 3.020443185172863, 3.651836888154007, 3.955596757145412, 4.800952752598403, 5.197004172516070, 5.971890284878268, 6.105543104885243, 6.909643929291096, 7.389627717874232, 7.838991221373860, 8.301848154150133, 9.096052512656010, 9.451655452775486, 9.916850756434929, 10.59174104215288, 10.98265783514805, 11.33341414772475, 11.94576343738245, 12.48701948760310, 12.72087861214193, 13.16783879904174, 13.93994787619157
