L(s) = 1 | − 9·3-s − 55.5·5-s − 2.50·7-s + 81·9-s − 228.·11-s + 658.·13-s + 499.·15-s − 1.44e3·17-s + 982.·19-s + 22.5·21-s − 529·23-s − 42.8·25-s − 729·27-s − 7.15e3·29-s + 9.25e3·31-s + 2.05e3·33-s + 139.·35-s + 2.42e3·37-s − 5.92e3·39-s − 4.07e3·41-s + 1.04e4·43-s − 4.49e3·45-s − 9.35e3·47-s − 1.68e4·49-s + 1.29e4·51-s − 3.42e4·53-s + 1.26e4·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.993·5-s − 0.0193·7-s + 0.333·9-s − 0.569·11-s + 1.08·13-s + 0.573·15-s − 1.21·17-s + 0.624·19-s + 0.0111·21-s − 0.208·23-s − 0.0137·25-s − 0.192·27-s − 1.58·29-s + 1.73·31-s + 0.328·33-s + 0.0191·35-s + 0.290·37-s − 0.624·39-s − 0.378·41-s + 0.859·43-s − 0.331·45-s − 0.617·47-s − 0.999·49-s + 0.699·51-s − 1.67·53-s + 0.565·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.7515749055\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7515749055\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
good | 5 | \( 1 + 55.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 2.50T + 1.68e4T^{2} \) |
| 11 | \( 1 + 228.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 658.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.44e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 982.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 7.15e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.25e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.42e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.07e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.04e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.35e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.42e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.26e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.34e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.39e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.11e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.00e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.61e4T + 8.58e9T^{2} \) |
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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
−9.068034817837854252413760497725, −8.166070466118250460827083245139, −7.55048989378911556948159511763, −6.56739369619575275151812773313, −5.80607700379345855233860528665, −4.71862239829514296638613682037, −4.00075624164073641631254321216, −2.99895821356065109551796385191, −1.60305724514617324391202612185, −0.38656200442501889003190434393,
0.38656200442501889003190434393, 1.60305724514617324391202612185, 2.99895821356065109551796385191, 4.00075624164073641631254321216, 4.71862239829514296638613682037, 5.80607700379345855233860528665, 6.56739369619575275151812773313, 7.55048989378911556948159511763, 8.166070466118250460827083245139, 9.068034817837854252413760497725