| L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s − 2·11-s + 2·13-s + 15-s + 4·21-s + 6·23-s + 25-s + 27-s − 8·29-s − 31-s − 2·33-s + 4·35-s − 2·37-s + 2·39-s + 6·41-s − 4·43-s + 45-s + 12·47-s + 9·49-s − 2·53-s − 2·55-s + 12·59-s − 8·61-s + 4·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.258·15-s + 0.872·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s − 0.179·31-s − 0.348·33-s + 0.676·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s + 1.75·47-s + 9/7·49-s − 0.274·53-s − 0.269·55-s + 1.56·59-s − 1.02·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.651685651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.651685651\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 31 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−7.83712248826493131187615567446, −7.44712941898896512136688416287, −6.60400770506928135280069672615, −5.54005383384359907134887755926, −5.19567794443771804849304678849, −4.33278575036573689680287218625, −3.55185762549580308047168023091, −2.53123061130001531890197266968, −1.87253803290458407246424528014, −0.994538679216161087291978332008,
0.994538679216161087291978332008, 1.87253803290458407246424528014, 2.53123061130001531890197266968, 3.55185762549580308047168023091, 4.33278575036573689680287218625, 5.19567794443771804849304678849, 5.54005383384359907134887755926, 6.60400770506928135280069672615, 7.44712941898896512136688416287, 7.83712248826493131187615567446
