| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 5·11-s + 5·13-s + 15-s + 2·17-s + 19-s − 21-s − 8·23-s − 4·25-s + 27-s + 9·29-s + 4·31-s + 5·33-s − 35-s + 5·39-s − 8·41-s + 45-s − 9·47-s − 6·49-s + 2·51-s + 5·55-s + 57-s − 4·59-s + 4·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.50·11-s + 1.38·13-s + 0.258·15-s + 0.485·17-s + 0.229·19-s − 0.218·21-s − 1.66·23-s − 4/5·25-s + 0.192·27-s + 1.67·29-s + 0.718·31-s + 0.870·33-s − 0.169·35-s + 0.800·39-s − 1.24·41-s + 0.149·45-s − 1.31·47-s − 6/7·49-s + 0.280·51-s + 0.674·55-s + 0.132·57-s − 0.520·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−12.79946925829916, −12.19882333311848, −11.93801880759913, −11.51444813926077, −10.94785823474428, −10.37533382471158, −9.821937847643272, −9.660194492953011, −9.278194964634351, −8.465767037609035, −8.215984366426737, −8.071384329239010, −7.103059258254774, −6.596469607662255, −6.244468019241018, −6.115337295287209, −5.203988993142890, −4.773726356816391, −3.949933169414171, −3.677713077571539, −3.410294252688505, −2.538219424600108, −2.043937497484219, −1.262415141269194, −1.145008994926330, 0,
1.145008994926330, 1.262415141269194, 2.043937497484219, 2.538219424600108, 3.410294252688505, 3.677713077571539, 3.949933169414171, 4.773726356816391, 5.203988993142890, 6.115337295287209, 6.244468019241018, 6.596469607662255, 7.103059258254774, 8.071384329239010, 8.215984366426737, 8.465767037609035, 9.278194964634351, 9.660194492953011, 9.821937847643272, 10.37533382471158, 10.94785823474428, 11.51444813926077, 11.93801880759913, 12.19882333311848, 12.79946925829916
