| L(s) = 1 | + 5.42·2-s + 3·3-s + 21.4·4-s + 16.2·6-s + 15.6·7-s + 73.1·8-s + 9·9-s − 11.2·11-s + 64.4·12-s + 70.4·13-s + 85.1·14-s + 225.·16-s − 117.·17-s + 48.8·18-s + 133.·19-s + 47.0·21-s − 61.0·22-s + 4.39·23-s + 219.·24-s + 382.·26-s + 27·27-s + 336.·28-s + 17.8·29-s − 179.·31-s + 638.·32-s − 33.7·33-s − 640.·34-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 0.577·3-s + 2.68·4-s + 1.10·6-s + 0.847·7-s + 3.23·8-s + 0.333·9-s − 0.308·11-s + 1.54·12-s + 1.50·13-s + 1.62·14-s + 3.52·16-s − 1.68·17-s + 0.639·18-s + 1.60·19-s + 0.489·21-s − 0.591·22-s + 0.0398·23-s + 1.86·24-s + 2.88·26-s + 0.192·27-s + 2.27·28-s + 0.114·29-s − 1.03·31-s + 3.52·32-s − 0.177·33-s − 3.22·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.43956996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.43956996\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 5.42T + 8T^{2} \) |
| 7 | \( 1 - 15.6T + 343T^{2} \) |
| 11 | \( 1 + 11.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 70.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 133.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 4.39T + 1.21e4T^{2} \) |
| 29 | \( 1 - 17.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 179.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 86.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 375.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 400.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 456.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 124.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 275.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 499.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 635.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 812.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.12e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 479.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 942.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 461.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.27e3T + 9.12e5T^{2} \) |
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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
−8.566101285241332374725360998190, −7.948308398271743930578836046015, −6.96385685793433529498545857459, −6.40841249893307809360375980585, −5.30228256458564753406306359265, −4.85314386681830632577847812015, −3.81629572125783679506681723714, −3.28901134913931001839007521845, −2.16517133119665129956777218359, −1.41827285446963880514390867467,
1.41827285446963880514390867467, 2.16517133119665129956777218359, 3.28901134913931001839007521845, 3.81629572125783679506681723714, 4.85314386681830632577847812015, 5.30228256458564753406306359265, 6.40841249893307809360375980585, 6.96385685793433529498545857459, 7.948308398271743930578836046015, 8.566101285241332374725360998190
