| L(s) = 1 | + 2·2-s + 4·4-s − 3.89·5-s + 8·8-s − 7.79·10-s + 61.3·11-s − 53.6·13-s + 16·16-s + 32.1·17-s + 55.7·19-s − 15.5·20-s + 122.·22-s + 94.6·23-s − 109.·25-s − 107.·26-s − 138.·29-s − 132.·31-s + 32·32-s + 64.2·34-s + 149.·37-s + 111.·38-s − 31.1·40-s + 427.·41-s + 437.·43-s + 245.·44-s + 189.·46-s + 57.0·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.348·5-s + 0.353·8-s − 0.246·10-s + 1.68·11-s − 1.14·13-s + 0.250·16-s + 0.457·17-s + 0.673·19-s − 0.174·20-s + 1.18·22-s + 0.857·23-s − 0.878·25-s − 0.810·26-s − 0.884·29-s − 0.768·31-s + 0.176·32-s + 0.323·34-s + 0.662·37-s + 0.476·38-s − 0.123·40-s + 1.62·41-s + 1.55·43-s + 0.841·44-s + 0.606·46-s + 0.176·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.397216935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.397216935\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.89T + 125T^{2} \) |
| 11 | \( 1 - 61.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 53.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 32.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 55.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 94.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 427.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 437.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 57.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 263.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 451.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 579.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 309.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.19e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.31e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 233.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.60e3T + 9.12e5T^{2} \) |
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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
−9.556553922645206099061499193762, −9.178946293394297247613449360882, −7.69946421238976340555733443791, −7.23695780485060622098073355405, −6.19498659143027945997095611510, −5.32004243077272155792473801965, −4.24630351917497696373809426088, −3.55790985314990327580576680415, −2.29872978431818329262918064618, −0.944137019811539919812770333919,
0.944137019811539919812770333919, 2.29872978431818329262918064618, 3.55790985314990327580576680415, 4.24630351917497696373809426088, 5.32004243077272155792473801965, 6.19498659143027945997095611510, 7.23695780485060622098073355405, 7.69946421238976340555733443791, 9.178946293394297247613449360882, 9.556553922645206099061499193762
