| L(s) = 1 | − 2.93·3-s − 2.06·5-s + 2.42·7-s + 5.61·9-s − 2.57·11-s − 1.28·13-s + 6.06·15-s + 6.19·17-s + 1.69·19-s − 7.12·21-s + 6.28·23-s − 0.723·25-s − 7.67·27-s + 5.09·29-s + 5.11·31-s + 7.56·33-s − 5.01·35-s − 10.0·37-s + 3.76·39-s − 0.199·41-s + 8.26·43-s − 11.6·45-s − 5.60·47-s − 1.11·49-s − 18.1·51-s − 3.03·53-s + 5.32·55-s + ⋯ |
| L(s) = 1 | − 1.69·3-s − 0.924·5-s + 0.916·7-s + 1.87·9-s − 0.776·11-s − 0.355·13-s + 1.56·15-s + 1.50·17-s + 0.389·19-s − 1.55·21-s + 1.30·23-s − 0.144·25-s − 1.47·27-s + 0.945·29-s + 0.919·31-s + 1.31·33-s − 0.848·35-s − 1.65·37-s + 0.603·39-s − 0.0312·41-s + 1.26·43-s − 1.73·45-s − 0.818·47-s − 0.159·49-s − 2.54·51-s − 0.416·53-s + 0.718·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8821592949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8821592949\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 2.93T + 3T^{2} \) |
| 5 | \( 1 + 2.06T + 5T^{2} \) |
| 7 | \( 1 - 2.42T + 7T^{2} \) |
| 11 | \( 1 + 2.57T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 - 6.19T + 17T^{2} \) |
| 19 | \( 1 - 1.69T + 19T^{2} \) |
| 23 | \( 1 - 6.28T + 23T^{2} \) |
| 29 | \( 1 - 5.09T + 29T^{2} \) |
| 31 | \( 1 - 5.11T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 0.199T + 41T^{2} \) |
| 43 | \( 1 - 8.26T + 43T^{2} \) |
| 47 | \( 1 + 5.60T + 47T^{2} \) |
| 53 | \( 1 + 3.03T + 53T^{2} \) |
| 59 | \( 1 - 3.92T + 59T^{2} \) |
| 61 | \( 1 - 5.50T + 61T^{2} \) |
| 67 | \( 1 - 5.34T + 67T^{2} \) |
| 71 | \( 1 + 4.73T + 71T^{2} \) |
| 73 | \( 1 - 4.46T + 73T^{2} \) |
| 79 | \( 1 + 0.736T + 79T^{2} \) |
| 83 | \( 1 + 7.29T + 83T^{2} \) |
| 89 | \( 1 - 3.85T + 89T^{2} \) |
| 97 | \( 1 + 1.69T + 97T^{2} \) |
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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
−7.88687172361859011086168414396, −7.35114816075749393510902487651, −6.69996798069633775118120647514, −5.75327196411627644244805181331, −5.05131187212680927935199422174, −4.87808368970505042981539398484, −3.86463450700262065946112985257, −2.86755452457420803461600232377, −1.40475737229903041698689472636, −0.59619346117953613702038110443,
0.59619346117953613702038110443, 1.40475737229903041698689472636, 2.86755452457420803461600232377, 3.86463450700262065946112985257, 4.87808368970505042981539398484, 5.05131187212680927935199422174, 5.75327196411627644244805181331, 6.69996798069633775118120647514, 7.35114816075749393510902487651, 7.88687172361859011086168414396
