| L(s) = 1 | + 3-s + 9-s − 6·11-s + 3·13-s − 4·17-s − 19-s − 4·23-s + 27-s + 8·29-s + 31-s − 6·33-s + 7·37-s + 3·39-s + 6·41-s − 43-s − 2·47-s − 4·51-s + 4·53-s − 57-s + 8·59-s − 14·61-s − 7·67-s − 4·69-s − 6·71-s + 73-s + 79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.80·11-s + 0.832·13-s − 0.970·17-s − 0.229·19-s − 0.834·23-s + 0.192·27-s + 1.48·29-s + 0.179·31-s − 1.04·33-s + 1.15·37-s + 0.480·39-s + 0.937·41-s − 0.152·43-s − 0.291·47-s − 0.560·51-s + 0.549·53-s − 0.132·57-s + 1.04·59-s − 1.79·61-s − 0.855·67-s − 0.481·69-s − 0.712·71-s + 0.117·73-s + 0.112·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.155678613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.155678613\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.92089351512995, −12.65397501480807, −12.02199133760259, −11.46551244810143, −10.96428498982068, −10.53371530482412, −10.16452891103502, −9.756874592905403, −8.911685996982957, −8.792661290218423, −8.129125915423062, −7.821641599025326, −7.423473057759291, −6.675694232944729, −6.196602044038663, −5.817946757866229, −5.099973136549214, −4.539973444894087, −4.249014443228418, −3.482006739770498, −2.875901345862729, −2.482785659435508, −1.996428685119569, −1.150079625928663, −0.4002440459687445,
0.4002440459687445, 1.150079625928663, 1.996428685119569, 2.482785659435508, 2.875901345862729, 3.482006739770498, 4.249014443228418, 4.539973444894087, 5.099973136549214, 5.817946757866229, 6.196602044038663, 6.675694232944729, 7.423473057759291, 7.821641599025326, 8.129125915423062, 8.792661290218423, 8.911685996982957, 9.756874592905403, 10.16452891103502, 10.53371530482412, 10.96428498982068, 11.46551244810143, 12.02199133760259, 12.65397501480807, 12.92089351512995
