| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 1.61·11-s − 6.31·13-s + 15-s + 7.92·17-s + 2.38·19-s + 21-s − 6.70·23-s + 25-s + 27-s + 7.92·29-s + 9.08·31-s + 1.61·33-s + 35-s − 7.92·37-s − 6.31·39-s − 1.22·41-s − 5.92·43-s + 45-s + 6.70·47-s + 49-s + 7.92·51-s − 3.61·53-s + 1.61·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.486·11-s − 1.75·13-s + 0.258·15-s + 1.92·17-s + 0.547·19-s + 0.218·21-s − 1.39·23-s + 0.200·25-s + 0.192·27-s + 1.47·29-s + 1.63·31-s + 0.280·33-s + 0.169·35-s − 1.30·37-s − 1.01·39-s − 0.191·41-s − 0.903·43-s + 0.149·45-s + 0.977·47-s + 0.142·49-s + 1.10·51-s − 0.496·53-s + 0.217·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.791291568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.791291568\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 + 6.31T + 13T^{2} \) |
| 17 | \( 1 - 7.92T + 17T^{2} \) |
| 19 | \( 1 - 2.38T + 19T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 - 9.08T + 31T^{2} \) |
| 37 | \( 1 + 7.92T + 37T^{2} \) |
| 41 | \( 1 + 1.22T + 41T^{2} \) |
| 43 | \( 1 + 5.92T + 43T^{2} \) |
| 47 | \( 1 - 6.70T + 47T^{2} \) |
| 53 | \( 1 + 3.61T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 4.70T + 61T^{2} \) |
| 67 | \( 1 - 9.14T + 67T^{2} \) |
| 71 | \( 1 - 8.31T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 0.775T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 6.31T + 97T^{2} \) |
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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
−8.401344637191992818245326103991, −8.027855374526106957630169270772, −7.20143196901302264215837322074, −6.49730868088744997645602370875, −5.41863592558928724701791373655, −4.89793765789950635163084214943, −3.86539337759410968825241217135, −2.93202775593325574614251434587, −2.12769673139638200412536425514, −1.01703331692460735965931100563,
1.01703331692460735965931100563, 2.12769673139638200412536425514, 2.93202775593325574614251434587, 3.86539337759410968825241217135, 4.89793765789950635163084214943, 5.41863592558928724701791373655, 6.49730868088744997645602370875, 7.20143196901302264215837322074, 8.027855374526106957630169270772, 8.401344637191992818245326103991
