| L(s) = 1 | + 3·11-s − 4·13-s − 3·17-s − 4·19-s + 2·23-s + 2·29-s − 4·31-s + 4·37-s − 10·41-s + 7·43-s − 4·47-s − 7·49-s − 8·53-s − 59-s − 4·61-s − 12·67-s − 14·71-s + 2·73-s + 83-s + 89-s − 17·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.904·11-s − 1.10·13-s − 0.727·17-s − 0.917·19-s + 0.417·23-s + 0.371·29-s − 0.718·31-s + 0.657·37-s − 1.56·41-s + 1.06·43-s − 0.583·47-s − 49-s − 1.09·53-s − 0.130·59-s − 0.512·61-s − 1.46·67-s − 1.66·71-s + 0.234·73-s + 0.109·83-s + 0.105·89-s − 1.72·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7179034761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7179034761\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−13.46659171090799, −13.02277220681560, −12.39398182171097, −12.19686733923304, −11.53160374090643, −11.10950369037281, −10.64194028508001, −10.08150237830217, −9.522488524044616, −9.160553188214545, −8.671427345479202, −8.116090054403785, −7.525788511741926, −7.036861912249325, −6.476164145252758, −6.205461156924045, −5.380063895522247, −4.834563489360009, −4.362325593716241, −3.904161882953042, −3.044812694621719, −2.654394790558236, −1.794409603348124, −1.404361095706824, −0.2458306018552652,
0.2458306018552652, 1.404361095706824, 1.794409603348124, 2.654394790558236, 3.044812694621719, 3.904161882953042, 4.362325593716241, 4.834563489360009, 5.380063895522247, 6.205461156924045, 6.476164145252758, 7.036861912249325, 7.525788511741926, 8.116090054403785, 8.671427345479202, 9.160553188214545, 9.522488524044616, 10.08150237830217, 10.64194028508001, 11.10950369037281, 11.53160374090643, 12.19686733923304, 12.39398182171097, 13.02277220681560, 13.46659171090799
