| L(s) = 1 | + 2.66·2-s + 5.12·4-s + 2.89·5-s − 0.268·7-s + 8.34·8-s + 7.72·10-s − 5.35·11-s + 3.85·13-s − 0.716·14-s + 12.0·16-s + 3.41·17-s − 4.28·19-s + 14.8·20-s − 14.2·22-s + 23-s + 3.38·25-s + 10.2·26-s − 1.37·28-s + 29-s + 3.06·31-s + 15.4·32-s + 9.10·34-s − 0.776·35-s + 5.65·37-s − 11.4·38-s + 24.1·40-s + 2.62·41-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 2.56·4-s + 1.29·5-s − 0.101·7-s + 2.95·8-s + 2.44·10-s − 1.61·11-s + 1.06·13-s − 0.191·14-s + 3.00·16-s + 0.827·17-s − 0.983·19-s + 3.31·20-s − 3.04·22-s + 0.208·23-s + 0.676·25-s + 2.01·26-s − 0.259·28-s + 0.185·29-s + 0.549·31-s + 2.72·32-s + 1.56·34-s − 0.131·35-s + 0.930·37-s − 1.85·38-s + 3.82·40-s + 0.410·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.029845634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.029845634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 5 | \( 1 - 2.89T + 5T^{2} \) |
| 7 | \( 1 + 0.268T + 7T^{2} \) |
| 11 | \( 1 + 5.35T + 11T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 19 | \( 1 + 4.28T + 19T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 - 2.62T + 41T^{2} \) |
| 43 | \( 1 + 6.20T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 9.30T + 59T^{2} \) |
| 61 | \( 1 - 9.39T + 61T^{2} \) |
| 67 | \( 1 + 5.45T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + 6.27T + 73T^{2} \) |
| 79 | \( 1 + 9.39T + 79T^{2} \) |
| 83 | \( 1 + 4.93T + 83T^{2} \) |
| 89 | \( 1 + 6.26T + 89T^{2} \) |
| 97 | \( 1 + 3.54T + 97T^{2} \) |
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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
−7.86047577635977413861166311283, −7.00654136752426995505637259302, −6.25848017312149670203439793957, −5.75776473986018503461184445018, −5.35505005032454401756245057640, −4.54863903886969367434075567942, −3.72548492974720713247227836483, −2.75585016029031403450158125689, −2.37638058194178123833138858440, −1.32807110849973462095066546763,
1.32807110849973462095066546763, 2.37638058194178123833138858440, 2.75585016029031403450158125689, 3.72548492974720713247227836483, 4.54863903886969367434075567942, 5.35505005032454401756245057640, 5.75776473986018503461184445018, 6.25848017312149670203439793957, 7.00654136752426995505637259302, 7.86047577635977413861166311283
