| L(s) = 1 | + 2-s + 1.66·3-s + 4-s − 4.15·5-s + 1.66·6-s − 2.52·7-s + 8-s − 0.216·9-s − 4.15·10-s + 3.26·11-s + 1.66·12-s − 3.39·13-s − 2.52·14-s − 6.92·15-s + 16-s − 4.86·17-s − 0.216·18-s − 2.71·19-s − 4.15·20-s − 4.21·21-s + 3.26·22-s + 23-s + 1.66·24-s + 12.2·25-s − 3.39·26-s − 5.36·27-s − 2.52·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.963·3-s + 0.5·4-s − 1.85·5-s + 0.681·6-s − 0.954·7-s + 0.353·8-s − 0.0721·9-s − 1.31·10-s + 0.983·11-s + 0.481·12-s − 0.942·13-s − 0.675·14-s − 1.78·15-s + 0.250·16-s − 1.18·17-s − 0.0509·18-s − 0.623·19-s − 0.928·20-s − 0.919·21-s + 0.695·22-s + 0.208·23-s + 0.340·24-s + 2.44·25-s − 0.666·26-s − 1.03·27-s − 0.477·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.148163150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.148163150\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 1.66T + 3T^{2} \) |
| 5 | \( 1 + 4.15T + 5T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 11 | \( 1 - 3.26T + 11T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 17 | \( 1 + 4.86T + 17T^{2} \) |
| 19 | \( 1 + 2.71T + 19T^{2} \) |
| 29 | \( 1 - 8.83T + 29T^{2} \) |
| 31 | \( 1 - 8.56T + 31T^{2} \) |
| 37 | \( 1 - 3.94T + 37T^{2} \) |
| 41 | \( 1 - 6.11T + 41T^{2} \) |
| 43 | \( 1 + 0.170T + 43T^{2} \) |
| 47 | \( 1 + 5.33T + 47T^{2} \) |
| 53 | \( 1 - 4.95T + 53T^{2} \) |
| 59 | \( 1 + 3.39T + 59T^{2} \) |
| 61 | \( 1 + 0.678T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 + 9.49T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−8.107339676532235363578768271867, −7.34220158183594277926614999911, −6.68316950759343936987403933322, −6.22614052704941916128690610174, −4.63966133705451770389115177398, −4.43638221973585116471024474188, −3.57896505022603588366507469528, −3.01087535552115828077734614161, −2.36354709246970369121131491637, −0.62701144793523842341379248826,
0.62701144793523842341379248826, 2.36354709246970369121131491637, 3.01087535552115828077734614161, 3.57896505022603588366507469528, 4.43638221973585116471024474188, 4.63966133705451770389115177398, 6.22614052704941916128690610174, 6.68316950759343936987403933322, 7.34220158183594277926614999911, 8.107339676532235363578768271867
