| L(s) = 1 | + 2.34·2-s + 3.51·4-s + 4.85·7-s + 3.56·8-s − 1.93·11-s + 2·13-s + 11.4·14-s + 1.34·16-s + 2.43·17-s + 19-s − 4.54·22-s − 4.36·23-s + 4.69·26-s + 17.0·28-s + 8.26·29-s − 1.17·31-s − 3.98·32-s + 5.71·34-s + 7.03·37-s + 2.34·38-s + 7.93·41-s − 3.71·43-s − 6.80·44-s − 10.2·46-s − 13.2·47-s + 16.6·49-s + 7.03·52-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.75·4-s + 1.83·7-s + 1.26·8-s − 0.583·11-s + 0.554·13-s + 3.05·14-s + 0.335·16-s + 0.590·17-s + 0.229·19-s − 0.968·22-s − 0.910·23-s + 0.921·26-s + 3.23·28-s + 1.53·29-s − 0.211·31-s − 0.703·32-s + 0.980·34-s + 1.15·37-s + 0.381·38-s + 1.23·41-s − 0.567·43-s − 1.02·44-s − 1.51·46-s − 1.93·47-s + 2.37·49-s + 0.975·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.643776935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.643776935\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.34T + 2T^{2} \) |
| 7 | \( 1 - 4.85T + 7T^{2} \) |
| 11 | \( 1 + 1.93T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 2.43T + 17T^{2} \) |
| 23 | \( 1 + 4.36T + 23T^{2} \) |
| 29 | \( 1 - 8.26T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 - 7.03T + 37T^{2} \) |
| 41 | \( 1 - 7.93T + 41T^{2} \) |
| 43 | \( 1 + 3.71T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 1.13T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 - 4.36T + 61T^{2} \) |
| 67 | \( 1 + 2.54T + 67T^{2} \) |
| 71 | \( 1 + 6.50T + 71T^{2} \) |
| 73 | \( 1 + 2.71T + 73T^{2} \) |
| 79 | \( 1 - 8.21T + 79T^{2} \) |
| 83 | \( 1 + 6.80T + 83T^{2} \) |
| 89 | \( 1 - 1.13T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−7.975543871811192703983321623335, −7.77236816506646628459913158568, −6.64852776202110500439357182450, −5.89766282549838383912421669310, −5.25317284637200083988533642359, −4.65691050375251754103758767409, −4.09101191376281501310767810946, −3.07987431289750515395022940578, −2.22969354577418503571933055177, −1.28706250173011800091311498749,
1.28706250173011800091311498749, 2.22969354577418503571933055177, 3.07987431289750515395022940578, 4.09101191376281501310767810946, 4.65691050375251754103758767409, 5.25317284637200083988533642359, 5.89766282549838383912421669310, 6.64852776202110500439357182450, 7.77236816506646628459913158568, 7.975543871811192703983321623335
