L(s) = 1 | − 2-s + 3-s − 5-s − 6-s + 8-s + 10-s − 4·11-s − 4·13-s − 15-s − 16-s − 2·17-s + 4·19-s + 4·22-s + 8·23-s + 24-s + 4·26-s − 27-s + 12·29-s + 30-s + 8·31-s − 4·33-s + 2·34-s + 2·37-s − 4·38-s − 4·39-s − 40-s + 4·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 0.316·10-s − 1.20·11-s − 1.10·13-s − 0.258·15-s − 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.852·22-s + 1.66·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 2.22·29-s + 0.182·30-s + 1.43·31-s − 0.696·33-s + 0.342·34-s + 0.328·37-s − 0.648·38-s − 0.640·39-s − 0.158·40-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.205833642\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.205833642\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
good | 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.624555245572691207547369531836, −9.367917707914361146225118573668, −8.771488972154385057655487978209, −8.545994369597774701034142300739, −8.106759564129112732155841772358, −7.78309140646253881266458062402, −7.55516461665085885235739113654, −6.98844568775064610371432386998, −6.47850207074394746548164556853, −6.39195419311599498297393451808, −5.19615608224426043602634059511, −5.17279028966263741706147033597, −4.73839018213162375176045640831, −4.37733798685066541506852847064, −3.32422285334760083178589690144, −3.24354806167984956244132997803, −2.56115017801408889501986885816, −2.24827803507575228709301800556, −1.19622712247552016307997896447, −0.53129358613126882268842931066,
0.53129358613126882268842931066, 1.19622712247552016307997896447, 2.24827803507575228709301800556, 2.56115017801408889501986885816, 3.24354806167984956244132997803, 3.32422285334760083178589690144, 4.37733798685066541506852847064, 4.73839018213162375176045640831, 5.17279028966263741706147033597, 5.19615608224426043602634059511, 6.39195419311599498297393451808, 6.47850207074394746548164556853, 6.98844568775064610371432386998, 7.55516461665085885235739113654, 7.78309140646253881266458062402, 8.106759564129112732155841772358, 8.545994369597774701034142300739, 8.771488972154385057655487978209, 9.367917707914361146225118573668, 9.624555245572691207547369531836