| L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s − 2·11-s + 3·13-s + 15-s − 3·17-s − 7·19-s − 2·21-s + 6·23-s − 4·25-s + 27-s + 7·31-s − 2·33-s − 2·35-s − 10·37-s + 3·39-s + 41-s + 12·43-s + 45-s + 12·47-s − 3·49-s − 3·51-s − 10·53-s − 2·55-s − 7·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.832·13-s + 0.258·15-s − 0.727·17-s − 1.60·19-s − 0.436·21-s + 1.25·23-s − 4/5·25-s + 0.192·27-s + 1.25·31-s − 0.348·33-s − 0.338·35-s − 1.64·37-s + 0.480·39-s + 0.156·41-s + 1.82·43-s + 0.149·45-s + 1.75·47-s − 3/7·49-s − 0.420·51-s − 1.37·53-s − 0.269·55-s − 0.927·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−7.56134107122285343819545911418, −6.64863214009416212147725763442, −6.30406782076837710869113135349, −5.49416322052646269151087457128, −4.52386131840929325198728912357, −3.90992025076914045747312212320, −2.95072815370241313820548435937, −2.40356793062540595163714808335, −1.40227572524070738114784779782, 0,
1.40227572524070738114784779782, 2.40356793062540595163714808335, 2.95072815370241313820548435937, 3.90992025076914045747312212320, 4.52386131840929325198728912357, 5.49416322052646269151087457128, 6.30406782076837710869113135349, 6.64863214009416212147725763442, 7.56134107122285343819545911418
