| L(s) = 1 | + 2·3-s − 2·4-s + 7-s + 9-s + 3·11-s − 4·12-s + 4·13-s + 4·16-s + 3·17-s + 19-s + 2·21-s − 4·27-s − 2·28-s + 6·29-s − 4·31-s + 6·33-s − 2·36-s − 2·37-s + 8·39-s − 6·41-s + 43-s − 6·44-s + 3·47-s + 8·48-s − 6·49-s + 6·51-s − 8·52-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s + 0.377·7-s + 1/3·9-s + 0.904·11-s − 1.15·12-s + 1.10·13-s + 16-s + 0.727·17-s + 0.229·19-s + 0.436·21-s − 0.769·27-s − 0.377·28-s + 1.11·29-s − 0.718·31-s + 1.04·33-s − 1/3·36-s − 0.328·37-s + 1.28·39-s − 0.937·41-s + 0.152·43-s − 0.904·44-s + 0.437·47-s + 1.15·48-s − 6/7·49-s + 0.840·51-s − 1.10·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.824309118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.824309118\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−10.91480268940863095584089699098, −9.774569434898319390985970077664, −9.093042453849772655367942297608, −8.415277882855381679327851057927, −7.79259928613141419667093336124, −6.36395749589543487340848898303, −5.15254746234400988386261721371, −3.94480584782122104516168200313, −3.23182893022898332662250292694, −1.43009506463758030263279983978,
1.43009506463758030263279983978, 3.23182893022898332662250292694, 3.94480584782122104516168200313, 5.15254746234400988386261721371, 6.36395749589543487340848898303, 7.79259928613141419667093336124, 8.415277882855381679327851057927, 9.093042453849772655367942297608, 9.774569434898319390985970077664, 10.91480268940863095584089699098
