L(s) = 1 | + 24.1·3-s − 179.·7-s + 339.·9-s + 653.·11-s − 284.·13-s + 383.·17-s + 2.56e3·19-s − 4.34e3·21-s − 948.·23-s + 2.33e3·27-s − 1.52e3·29-s − 3.10e3·31-s + 1.57e4·33-s + 9.99e3·37-s − 6.87e3·39-s + 1.51e4·41-s + 1.75e3·43-s + 1.47e4·47-s + 1.55e4·49-s + 9.26e3·51-s + 8.70e3·53-s + 6.18e4·57-s + 1.26e4·59-s + 4.30e4·61-s − 6.10e4·63-s − 2.61e4·67-s − 2.28e4·69-s + ⋯ |
L(s) = 1 | + 1.54·3-s − 1.38·7-s + 1.39·9-s + 1.62·11-s − 0.467·13-s + 0.321·17-s + 1.62·19-s − 2.14·21-s − 0.373·23-s + 0.615·27-s − 0.336·29-s − 0.580·31-s + 2.52·33-s + 1.19·37-s − 0.723·39-s + 1.40·41-s + 0.144·43-s + 0.974·47-s + 0.925·49-s + 0.498·51-s + 0.425·53-s + 2.52·57-s + 0.472·59-s + 1.48·61-s − 1.93·63-s − 0.711·67-s − 0.578·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(3.721551064\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.721551064\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 24.1T + 243T^{2} \) |
| 7 | \( 1 + 179.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 653.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 284.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 383.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.56e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 948.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.52e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.10e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.99e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.51e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.75e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.47e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.70e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.26e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.30e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.93e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.95e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.30e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.62e4T + 8.58e9T^{2} \) |
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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
−9.850872949466792098623225714743, −9.523122446142492930314643128365, −8.880477417830469797757309496289, −7.65555591718489364805645790493, −6.95181212045924099148501120438, −5.82135811472331235590618549486, −4.03646394534799393285478890966, −3.40334086674757362487191029172, −2.43512919351404711329535215460, −0.986581885097227113280976576603,
0.986581885097227113280976576603, 2.43512919351404711329535215460, 3.40334086674757362487191029172, 4.03646394534799393285478890966, 5.82135811472331235590618549486, 6.95181212045924099148501120438, 7.65555591718489364805645790493, 8.880477417830469797757309496289, 9.523122446142492930314643128365, 9.850872949466792098623225714743