| L(s) = 1 | − 4·11-s + 13-s + 2·17-s − 4·19-s − 8·23-s − 2·29-s + 8·31-s + 6·37-s + 6·41-s + 4·43-s + 8·47-s − 7·49-s − 6·53-s + 12·59-s + 2·61-s + 4·67-s + 6·73-s − 16·79-s − 4·83-s − 10·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 0.277·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s − 0.371·29-s + 1.43·31-s + 0.986·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s − 49-s − 0.824·53-s + 1.56·59-s + 0.256·61-s + 0.488·67-s + 0.702·73-s − 1.80·79-s − 0.439·83-s − 1.05·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.468936594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.468936594\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.02262195485945, −12.64987661973890, −12.38373635667454, −11.54666531891568, −11.35996054808506, −10.70764687185735, −10.28000002200442, −9.853990848646887, −9.516769124645556, −8.648631597422225, −8.349962692553619, −7.838235042656828, −7.560058292136114, −6.798073158442697, −6.247065859653020, −5.820449381389047, −5.404168291030624, −4.696142961900060, −4.138638914425153, −3.831023563686522, −2.831021830759423, −2.581196401835364, −1.940472554110014, −1.128232316497176, −0.3638917140422578,
0.3638917140422578, 1.128232316497176, 1.940472554110014, 2.581196401835364, 2.831021830759423, 3.831023563686522, 4.138638914425153, 4.696142961900060, 5.404168291030624, 5.820449381389047, 6.247065859653020, 6.798073158442697, 7.560058292136114, 7.838235042656828, 8.349962692553619, 8.648631597422225, 9.516769124645556, 9.853990848646887, 10.28000002200442, 10.70764687185735, 11.35996054808506, 11.54666531891568, 12.38373635667454, 12.64987661973890, 13.02262195485945
