L(s) = 1 | − 1.94·3-s − 1.68·5-s − 1.54·7-s + 0.784·9-s − 2.23·11-s − 1.94·13-s + 3.28·15-s − 2.28·17-s + 19-s + 3.00·21-s + 0.628·23-s − 2.14·25-s + 4.30·27-s + 9.85·29-s + 8.81·31-s + 4.34·33-s + 2.60·35-s − 2.52·37-s + 3.78·39-s + 8.82·41-s − 2.39·43-s − 1.32·45-s − 1.22·47-s − 4.61·49-s + 4.44·51-s + 5.98·53-s + 3.77·55-s + ⋯ |
L(s) = 1 | − 1.12·3-s − 0.755·5-s − 0.583·7-s + 0.261·9-s − 0.673·11-s − 0.540·13-s + 0.848·15-s − 0.554·17-s + 0.229·19-s + 0.655·21-s + 0.131·23-s − 0.429·25-s + 0.829·27-s + 1.82·29-s + 1.58·31-s + 0.756·33-s + 0.441·35-s − 0.415·37-s + 0.606·39-s + 1.37·41-s − 0.365·43-s − 0.197·45-s − 0.178·47-s − 0.659·49-s + 0.622·51-s + 0.822·53-s + 0.508·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 + 1.94T + 3T^{2} \) |
| 5 | \( 1 + 1.68T + 5T^{2} \) |
| 7 | \( 1 + 1.54T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 1.94T + 13T^{2} \) |
| 17 | \( 1 + 2.28T + 17T^{2} \) |
| 23 | \( 1 - 0.628T + 23T^{2} \) |
| 29 | \( 1 - 9.85T + 29T^{2} \) |
| 31 | \( 1 - 8.81T + 31T^{2} \) |
| 37 | \( 1 + 2.52T + 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 + 1.22T + 47T^{2} \) |
| 53 | \( 1 - 5.98T + 53T^{2} \) |
| 59 | \( 1 + 8.90T + 59T^{2} \) |
| 61 | \( 1 + 0.227T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 0.0896T + 89T^{2} \) |
| 97 | \( 1 + 6.86T + 97T^{2} \) |
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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
−7.75175867152268961584185978608, −6.77546109294162006474742763665, −6.41877133268097042976336298120, −5.57467421086692136079239036464, −4.82324989552928082245499709835, −4.31183031202181355836842097685, −3.17245560226099658308889935791, −2.48405525838619087878312605239, −0.877322514979659159441187585786, 0,
0.877322514979659159441187585786, 2.48405525838619087878312605239, 3.17245560226099658308889935791, 4.31183031202181355836842097685, 4.82324989552928082245499709835, 5.57467421086692136079239036464, 6.41877133268097042976336298120, 6.77546109294162006474742763665, 7.75175867152268961584185978608