| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 2·12-s − 13-s + 16-s + 2·17-s − 18-s + 4·19-s + 6·23-s + 2·24-s + 26-s + 4·27-s + 6·29-s − 32-s − 2·34-s + 36-s − 8·37-s − 4·38-s + 2·39-s − 10·41-s + 8·43-s − 6·46-s + 6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.577·12-s − 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 1.25·23-s + 0.408·24-s + 0.196·26-s + 0.769·27-s + 1.11·29-s − 0.176·32-s − 0.342·34-s + 1/6·36-s − 1.31·37-s − 0.648·38-s + 0.320·39-s − 1.56·41-s + 1.21·43-s − 0.884·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−14.28991049623468, −13.83794349220412, −13.11486508039040, −12.53828377466744, −12.11355242893184, −11.73076646946852, −11.26157662323872, −10.72871198211254, −10.35153059518095, −9.811718702790439, −9.343836124865788, −8.558710605162951, −8.389709465009713, −7.425487979618324, −7.127880852477992, −6.628258877623650, −5.972431113726924, −5.510309201278271, −4.954826468635193, −4.543491475597802, −3.397944547267792, −3.089421051632039, −2.226542242549076, −1.307088536486372, −0.8327832174442314, 0,
0.8327832174442314, 1.307088536486372, 2.226542242549076, 3.089421051632039, 3.397944547267792, 4.543491475597802, 4.954826468635193, 5.510309201278271, 5.972431113726924, 6.628258877623650, 7.127880852477992, 7.425487979618324, 8.389709465009713, 8.558710605162951, 9.343836124865788, 9.811718702790439, 10.35153059518095, 10.72871198211254, 11.26157662323872, 11.73076646946852, 12.11355242893184, 12.53828377466744, 13.11486508039040, 13.83794349220412, 14.28991049623468
