| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s − 13-s + 16-s − 3·17-s − 4·19-s − 2·20-s − 3·23-s − 25-s − 26-s + 7·29-s + 11·31-s + 32-s − 3·34-s − 6·37-s − 4·38-s − 2·40-s + 6·41-s − 43-s − 3·46-s − 8·47-s − 50-s − 52-s − 53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s − 0.277·13-s + 1/4·16-s − 0.727·17-s − 0.917·19-s − 0.447·20-s − 0.625·23-s − 1/5·25-s − 0.196·26-s + 1.29·29-s + 1.97·31-s + 0.176·32-s − 0.514·34-s − 0.986·37-s − 0.648·38-s − 0.316·40-s + 0.937·41-s − 0.152·43-s − 0.442·46-s − 1.16·47-s − 0.141·50-s − 0.138·52-s − 0.137·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.221786711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.221786711\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 11 T + p T^{2} \) | 1.71.al |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−12.45790352533637, −12.07473671052470, −11.98800378053171, −11.36297872699286, −10.83218908282950, −10.56533021033197, −9.912247787395974, −9.588523193426017, −8.749822550533521, −8.357511095795117, −8.073299357365692, −7.511040299773297, −6.925275118033416, −6.520608775803298, −6.153590668729770, −5.554575421540645, −4.783740948342621, −4.533267145731212, −4.136622885039510, −3.601007120325337, −2.852585092311776, −2.608212921171178, −1.830537017128374, −1.188397524093414, −0.2574993348035196,
0.2574993348035196, 1.188397524093414, 1.830537017128374, 2.608212921171178, 2.852585092311776, 3.601007120325337, 4.136622885039510, 4.533267145731212, 4.783740948342621, 5.554575421540645, 6.153590668729770, 6.520608775803298, 6.925275118033416, 7.511040299773297, 8.073299357365692, 8.357511095795117, 8.749822550533521, 9.588523193426017, 9.912247787395974, 10.56533021033197, 10.83218908282950, 11.36297872699286, 11.98800378053171, 12.07473671052470, 12.45790352533637
