| L(s) = 1 | + 3·3-s − 9.15·5-s + 27.4·7-s + 9·9-s + 20.5·11-s + 32.0·13-s − 27.4·15-s − 111.·17-s + 129.·19-s + 82.2·21-s + 9.16·23-s − 41.1·25-s + 27·27-s + 41.0·29-s + 187.·31-s + 61.5·33-s − 251.·35-s − 114.·37-s + 96.1·39-s − 282.·41-s − 89.3·43-s − 82.3·45-s + 54.6·47-s + 408.·49-s − 335.·51-s + 726.·53-s − 187.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.818·5-s + 1.48·7-s + 0.333·9-s + 0.562·11-s + 0.683·13-s − 0.472·15-s − 1.59·17-s + 1.56·19-s + 0.854·21-s + 0.0830·23-s − 0.329·25-s + 0.192·27-s + 0.262·29-s + 1.08·31-s + 0.324·33-s − 1.21·35-s − 0.507·37-s + 0.394·39-s − 1.07·41-s − 0.317·43-s − 0.272·45-s + 0.169·47-s + 1.19·49-s − 0.920·51-s + 1.88·53-s − 0.460·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.832755353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.832755353\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| good | 5 | \( 1 + 9.15T + 125T^{2} \) |
| 7 | \( 1 - 27.4T + 343T^{2} \) |
| 11 | \( 1 - 20.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 9.16T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 187.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 89.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 54.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 726.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 216.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 754.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 379.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 302.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 504.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 301.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 599.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 277.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 765.T + 9.12e5T^{2} \) |
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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
−9.870269870956191140442527107333, −8.729860632018858142053937911475, −8.361399291612827157870277437647, −7.49433752466680419444691874723, −6.66256512860488179193790296043, −5.23092047371621572359286147171, −4.35961384888422383371793468318, −3.54620461773944231609677831139, −2.13403003213766129772082610183, −0.987458542365775112039894632395,
0.987458542365775112039894632395, 2.13403003213766129772082610183, 3.54620461773944231609677831139, 4.35961384888422383371793468318, 5.23092047371621572359286147171, 6.66256512860488179193790296043, 7.49433752466680419444691874723, 8.361399291612827157870277437647, 8.729860632018858142053937911475, 9.870269870956191140442527107333
