| L(s) = 1 | − 3·3-s − 5.56·5-s + 9·9-s + 13.9·11-s − 38.6·13-s + 16.6·15-s − 43.4·17-s − 109.·19-s + 74.8·23-s − 94.0·25-s − 27·27-s − 72.3·29-s + 64.0·31-s − 41.7·33-s + 188.·37-s + 116.·39-s + 24.7·41-s + 243.·43-s − 50.0·45-s − 620.·47-s + 130.·51-s − 287.·53-s − 77.4·55-s + 327.·57-s + 525.·59-s + 383.·61-s + 215.·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.497·5-s + 0.333·9-s + 0.381·11-s − 0.825·13-s + 0.287·15-s − 0.620·17-s − 1.31·19-s + 0.678·23-s − 0.752·25-s − 0.192·27-s − 0.463·29-s + 0.371·31-s − 0.220·33-s + 0.838·37-s + 0.476·39-s + 0.0944·41-s + 0.864·43-s − 0.165·45-s − 1.92·47-s + 0.358·51-s − 0.745·53-s − 0.189·55-s + 0.760·57-s + 1.15·59-s + 0.804·61-s + 0.410·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8099819770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8099819770\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 5.56T + 125T^{2} \) |
| 11 | \( 1 - 13.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 43.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 109.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 74.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 72.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 64.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 188.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 24.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 243.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 620.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 287.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 525.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 383.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 198.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 785.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 331.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 437.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 241.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.58e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 79.2T + 9.12e5T^{2} \) |
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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
−8.575871093688020593463773863649, −7.83528122049696283700378295447, −6.98817990098804393914937681136, −6.39984552928613171418469899137, −5.49028917851218004532516612083, −4.53660999923034490280465659815, −4.03141998487483639560084226125, −2.77847761236670127862072395060, −1.73065962922463485367412973883, −0.40553889271913398338234579370,
0.40553889271913398338234579370, 1.73065962922463485367412973883, 2.77847761236670127862072395060, 4.03141998487483639560084226125, 4.53660999923034490280465659815, 5.49028917851218004532516612083, 6.39984552928613171418469899137, 6.98817990098804393914937681136, 7.83528122049696283700378295447, 8.575871093688020593463773863649
