| L(s) = 1 | + 3·3-s + 9.82·5-s + 9·9-s + 14.1·11-s − 26.1·13-s + 29.4·15-s − 78.5·17-s − 73.1·19-s + 96·23-s − 28.4·25-s + 27·27-s + 173.·29-s − 67.2·31-s + 42.5·33-s − 301.·37-s − 78.3·39-s + 472.·41-s + 463.·43-s + 88.4·45-s + 91.1·47-s − 235.·51-s − 163.·53-s + 139.·55-s − 219.·57-s + 600.·59-s + 571.·61-s − 256.·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.878·5-s + 0.333·9-s + 0.388·11-s − 0.557·13-s + 0.507·15-s − 1.12·17-s − 0.883·19-s + 0.870·23-s − 0.227·25-s + 0.192·27-s + 1.10·29-s − 0.389·31-s + 0.224·33-s − 1.34·37-s − 0.321·39-s + 1.79·41-s + 1.64·43-s + 0.292·45-s + 0.283·47-s − 0.647·51-s − 0.423·53-s + 0.341·55-s − 0.510·57-s + 1.32·59-s + 1.19·61-s − 0.489·65-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
| \(L(2)\) |
\(\approx\) |
\(3.335919731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.335919731\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 9.82T + 125T^{2} \) |
| 11 | \( 1 - 14.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 78.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 73.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 96T + 1.21e4T^{2} \) |
| 29 | \( 1 - 173.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 67.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 301.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 472.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 463.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 91.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 163.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 600.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 571.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 539.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 442.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 45.7T + 4.93e5T^{2} \) |
| 83 | \( 1 - 686.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 660.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 658.T + 9.12e5T^{2} \) |
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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
−8.874499298812786358990906812186, −7.938607024814203285967374030786, −6.97965011952350634833479983560, −6.45009285383483208726035701550, −5.49365689469031827991917415769, −4.59439271856733002744055599031, −3.78402675770221741379897159764, −2.52127845396631951456084097280, −2.07648689982123181363826727206, −0.78321511856135708691518449330,
0.78321511856135708691518449330, 2.07648689982123181363826727206, 2.52127845396631951456084097280, 3.78402675770221741379897159764, 4.59439271856733002744055599031, 5.49365689469031827991917415769, 6.45009285383483208726035701550, 6.97965011952350634833479983560, 7.938607024814203285967374030786, 8.874499298812786358990906812186
