| L(s) = 1 | + 3·5-s + 3·11-s + 6·13-s − 6·17-s − 2·19-s + 6·23-s + 4·25-s − 6·29-s + 3·31-s − 6·37-s + 6·41-s + 8·43-s − 12·47-s + 3·53-s + 9·55-s − 12·59-s + 18·65-s + 4·67-s − 6·71-s + 11·73-s + 12·79-s + 9·83-s − 18·85-s − 6·95-s + 17·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.904·11-s + 1.66·13-s − 1.45·17-s − 0.458·19-s + 1.25·23-s + 4/5·25-s − 1.11·29-s + 0.538·31-s − 0.986·37-s + 0.937·41-s + 1.21·43-s − 1.75·47-s + 0.412·53-s + 1.21·55-s − 1.56·59-s + 2.23·65-s + 0.488·67-s − 0.712·71-s + 1.28·73-s + 1.35·79-s + 0.987·83-s − 1.95·85-s − 0.615·95-s + 1.72·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.817291712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.817291712\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−12.80509800592263, −12.14510154292148, −11.55962120012014, −11.02523257194245, −10.84660387534991, −10.48233281576389, −9.645110951103692, −9.333898002442400, −8.995125845803355, −8.682465816091703, −8.095250029272744, −7.429631125843434, −6.762102338000807, −6.449909150897370, −6.162169656781504, −5.680494765881758, −5.045801925918348, −4.566429257895416, −3.963357423734775, −3.482599519247177, −2.896119127004252, −2.085627708712761, −1.844666926217134, −1.204070302844792, −0.5860734217286722,
0.5860734217286722, 1.204070302844792, 1.844666926217134, 2.085627708712761, 2.896119127004252, 3.482599519247177, 3.963357423734775, 4.566429257895416, 5.045801925918348, 5.680494765881758, 6.162169656781504, 6.449909150897370, 6.762102338000807, 7.429631125843434, 8.095250029272744, 8.682465816091703, 8.995125845803355, 9.333898002442400, 9.645110951103692, 10.48233281576389, 10.84660387534991, 11.02523257194245, 11.55962120012014, 12.14510154292148, 12.80509800592263
