| L(s) = 1 | + 3-s + 2·5-s − 7-s + 9-s − 4·13-s + 2·15-s − 6·19-s − 21-s − 23-s − 25-s + 27-s + 6·29-s − 10·31-s − 2·35-s + 6·37-s − 4·39-s − 2·41-s − 12·43-s + 2·45-s + 10·47-s + 49-s + 10·53-s − 6·57-s + 12·59-s + 14·61-s − 63-s − 8·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.10·13-s + 0.516·15-s − 1.37·19-s − 0.218·21-s − 0.208·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.79·31-s − 0.338·35-s + 0.986·37-s − 0.640·39-s − 0.312·41-s − 1.82·43-s + 0.298·45-s + 1.45·47-s + 1/7·49-s + 1.37·53-s − 0.794·57-s + 1.56·59-s + 1.79·61-s − 0.125·63-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.504252390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.504252390\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−15.02426440586059, −14.46699682209579, −14.19761431490920, −13.44188427704824, −13.00217846542231, −12.70585592551919, −11.96784973885546, −11.43434832217242, −10.57280271187522, −10.11711650861341, −9.819978464375436, −9.160498296987230, −8.649474828956863, −8.112147060475770, −7.369066789567620, −6.761271504549525, −6.398152873331626, −5.438880734046867, −5.200123124151118, −4.127237559129334, −3.810155389076563, −2.697963293714470, −2.334667099131178, −1.734517336988781, −0.5527496354067867,
0.5527496354067867, 1.734517336988781, 2.334667099131178, 2.697963293714470, 3.810155389076563, 4.127237559129334, 5.200123124151118, 5.438880734046867, 6.398152873331626, 6.761271504549525, 7.369066789567620, 8.112147060475770, 8.649474828956863, 9.160498296987230, 9.819978464375436, 10.11711650861341, 10.57280271187522, 11.43434832217242, 11.96784973885546, 12.70585592551919, 13.00217846542231, 13.44188427704824, 14.19761431490920, 14.46699682209579, 15.02426440586059
