| L(s) = 1 | − 3-s + 2.82·7-s + 9-s − 2.97·11-s + 13-s − 1.44·17-s + 4.17·19-s − 2.82·21-s − 6.21·23-s − 27-s − 0.828·29-s + 1.65·31-s + 2.97·33-s + 0.0418·37-s − 39-s − 4.36·41-s + 8.99·43-s + 4.40·47-s + 1.00·49-s + 1.44·51-s + 6.72·53-s − 4.17·57-s − 2.97·59-s + 6.27·61-s + 2.82·63-s − 0.727·67-s + 6.21·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.06·7-s + 0.333·9-s − 0.898·11-s + 0.277·13-s − 0.350·17-s + 0.956·19-s − 0.617·21-s − 1.29·23-s − 0.192·27-s − 0.153·29-s + 0.297·31-s + 0.518·33-s + 0.00688·37-s − 0.160·39-s − 0.681·41-s + 1.37·43-s + 0.642·47-s + 0.142·49-s + 0.202·51-s + 0.924·53-s − 0.552·57-s − 0.387·59-s + 0.803·61-s + 0.356·63-s − 0.0888·67-s + 0.747·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.674458024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.674458024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2.97T + 11T^{2} \) |
| 17 | \( 1 + 1.44T + 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 + 6.21T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 - 1.65T + 31T^{2} \) |
| 37 | \( 1 - 0.0418T + 37T^{2} \) |
| 41 | \( 1 + 4.36T + 41T^{2} \) |
| 43 | \( 1 - 8.99T + 43T^{2} \) |
| 47 | \( 1 - 4.40T + 47T^{2} \) |
| 53 | \( 1 - 6.72T + 53T^{2} \) |
| 59 | \( 1 + 2.97T + 59T^{2} \) |
| 61 | \( 1 - 6.27T + 61T^{2} \) |
| 67 | \( 1 + 0.727T + 67T^{2} \) |
| 71 | \( 1 + 8.32T + 71T^{2} \) |
| 73 | \( 1 - 6.87T + 73T^{2} \) |
| 79 | \( 1 - 6.11T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 6.78T + 97T^{2} \) |
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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
−7.70774619946857859095986150859, −7.37865462858473343377089119462, −6.34122857303469263462634083324, −5.68651302433080708384897741846, −5.10669100397354025692682665061, −4.45423879598615738266760145735, −3.65624354485832973876511367936, −2.53427001903886427552440171945, −1.73893526077584047457498087662, −0.67310170329487669446547646675,
0.67310170329487669446547646675, 1.73893526077584047457498087662, 2.53427001903886427552440171945, 3.65624354485832973876511367936, 4.45423879598615738266760145735, 5.10669100397354025692682665061, 5.68651302433080708384897741846, 6.34122857303469263462634083324, 7.37865462858473343377089119462, 7.70774619946857859095986150859
