| L(s) = 1 | + 3-s − 5-s − 2·9-s − 5·11-s + 2·13-s − 15-s − 3·17-s + 19-s − 6·23-s + 25-s − 5·27-s − 10·29-s − 31-s − 5·33-s − 5·37-s + 2·39-s + 6·41-s − 7·43-s + 2·45-s − 9·47-s − 3·51-s + 5·53-s + 5·55-s + 57-s + 10·59-s − 4·61-s − 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s − 1.50·11-s + 0.554·13-s − 0.258·15-s − 0.727·17-s + 0.229·19-s − 1.25·23-s + 1/5·25-s − 0.962·27-s − 1.85·29-s − 0.179·31-s − 0.870·33-s − 0.821·37-s + 0.320·39-s + 0.937·41-s − 1.06·43-s + 0.298·45-s − 1.31·47-s − 0.420·51-s + 0.686·53-s + 0.674·55-s + 0.132·57-s + 1.30·59-s − 0.512·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 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
−12.96137305471679, −12.58051319717467, −11.81982731462500, −11.54008840026769, −11.03912735286936, −10.70621762161218, −10.14558995482178, −9.622818517692031, −9.195798048165954, −8.582770673623780, −8.281811151378839, −7.885456362176840, −7.480987349867885, −6.934393981021661, −6.287255723773591, −5.775004554544517, −5.308268805003770, −4.925792398073003, −4.117209226640642, −3.650432347137679, −3.312724357351465, −2.585046745965621, −2.145297911901747, −1.679292439162684, −0.5251907555532684, 0,
0.5251907555532684, 1.679292439162684, 2.145297911901747, 2.585046745965621, 3.312724357351465, 3.650432347137679, 4.117209226640642, 4.925792398073003, 5.308268805003770, 5.775004554544517, 6.287255723773591, 6.934393981021661, 7.480987349867885, 7.885456362176840, 8.281811151378839, 8.582770673623780, 9.195798048165954, 9.622818517692031, 10.14558995482178, 10.70621762161218, 11.03912735286936, 11.54008840026769, 11.81982731462500, 12.58051319717467, 12.96137305471679
