| L(s) = 1 | + 3-s + 3·7-s + 9-s + 3·11-s − 13-s + 7·17-s + 8·19-s + 3·21-s + 4·23-s + 27-s + 3·29-s − 11·31-s + 3·33-s − 39-s − 2·41-s + 8·43-s − 9·47-s + 2·49-s + 7·51-s − 9·53-s + 8·57-s + 9·59-s − 61-s + 3·63-s − 5·67-s + 4·69-s − 12·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 1.69·17-s + 1.83·19-s + 0.654·21-s + 0.834·23-s + 0.192·27-s + 0.557·29-s − 1.97·31-s + 0.522·33-s − 0.160·39-s − 0.312·41-s + 1.21·43-s − 1.31·47-s + 2/7·49-s + 0.980·51-s − 1.23·53-s + 1.05·57-s + 1.17·59-s − 0.128·61-s + 0.377·63-s − 0.610·67-s + 0.481·69-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.262224479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.262224479\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 11 T + p T^{2} \) | 1.31.l |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.40219965579096, −13.97742630951008, −13.35227450456131, −12.71749101814986, −12.23442519738259, −11.67035722653115, −11.37969116984314, −10.76952029110886, −9.986795286297733, −9.730294286031976, −8.986210533311230, −8.786119896548591, −7.855238179246588, −7.582006326794913, −7.272978052989554, −6.435168754592060, −5.684960874680603, −5.158271208450149, −4.776853586058349, −3.922848095550217, −3.331495492583664, −2.964256258781803, −1.879140394979888, −1.404568737398945, −0.8256622566659866,
0.8256622566659866, 1.404568737398945, 1.879140394979888, 2.964256258781803, 3.331495492583664, 3.922848095550217, 4.776853586058349, 5.158271208450149, 5.684960874680603, 6.435168754592060, 7.272978052989554, 7.582006326794913, 7.855238179246588, 8.786119896548591, 8.986210533311230, 9.730294286031976, 9.986795286297733, 10.76952029110886, 11.37969116984314, 11.67035722653115, 12.23442519738259, 12.71749101814986, 13.35227450456131, 13.97742630951008, 14.40219965579096
