| L(s) = 1 | + 2-s − 4-s + 5·7-s − 3·8-s + 2·11-s + 5·14-s − 16-s − 4·17-s − 3·19-s + 2·22-s + 6·23-s − 5·28-s − 8·29-s + 31-s + 5·32-s − 4·34-s + 37-s − 3·38-s − 12·41-s − 43-s − 2·44-s + 6·46-s + 8·47-s + 18·49-s + 2·53-s − 15·56-s − 8·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.88·7-s − 1.06·8-s + 0.603·11-s + 1.33·14-s − 1/4·16-s − 0.970·17-s − 0.688·19-s + 0.426·22-s + 1.25·23-s − 0.944·28-s − 1.48·29-s + 0.179·31-s + 0.883·32-s − 0.685·34-s + 0.164·37-s − 0.486·38-s − 1.87·41-s − 0.152·43-s − 0.301·44-s + 0.884·46-s + 1.16·47-s + 18/7·49-s + 0.274·53-s − 2.00·56-s − 1.05·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.575497199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.575497199\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.54251907219366, −13.28770014803742, −12.73894395033964, −12.20695452433956, −11.55736797915882, −11.36485023978654, −10.95410753439516, −10.25524563773946, −9.696922406518002, −8.897167018589036, −8.715235233490660, −8.404439664713736, −7.640095849996224, −7.060985048599123, −6.618224768346240, −5.834662477817044, −5.273578308470028, −5.025354260762628, −4.351533133155879, −4.044016905885994, −3.461717828144464, −2.526223616904198, −2.014483539566251, −1.325855112413849, −0.5367697187877515,
0.5367697187877515, 1.325855112413849, 2.014483539566251, 2.526223616904198, 3.461717828144464, 4.044016905885994, 4.351533133155879, 5.025354260762628, 5.273578308470028, 5.834662477817044, 6.618224768346240, 7.060985048599123, 7.640095849996224, 8.404439664713736, 8.715235233490660, 8.897167018589036, 9.696922406518002, 10.25524563773946, 10.95410753439516, 11.36485023978654, 11.55736797915882, 12.20695452433956, 12.73894395033964, 13.28770014803742, 13.54251907219366
