| L(s) = 1 | + 0.263·3-s + 4.19·5-s − 1.25·7-s − 2.93·9-s + 1.03·11-s + 1.70·13-s + 1.10·15-s + 0.0209·17-s + 4.05·19-s − 0.331·21-s + 3.12·23-s + 12.6·25-s − 1.56·27-s − 5.01·29-s + 5.83·31-s + 0.273·33-s − 5.27·35-s − 6.49·37-s + 0.449·39-s + 10.1·41-s + 11.8·43-s − 12.3·45-s − 3.83·47-s − 5.41·49-s + 0.00553·51-s − 7.11·53-s + 4.35·55-s + ⋯ |
| L(s) = 1 | + 0.152·3-s + 1.87·5-s − 0.475·7-s − 0.976·9-s + 0.313·11-s + 0.472·13-s + 0.285·15-s + 0.00509·17-s + 0.931·19-s − 0.0723·21-s + 0.651·23-s + 2.52·25-s − 0.300·27-s − 0.931·29-s + 1.04·31-s + 0.0476·33-s − 0.892·35-s − 1.06·37-s + 0.0719·39-s + 1.57·41-s + 1.80·43-s − 1.83·45-s − 0.560·47-s − 0.774·49-s + 0.000774·51-s − 0.977·53-s + 0.587·55-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
| \(L(1)\) |
\(\approx\) |
\(3.024022687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.024022687\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.263T + 3T^{2} \) |
| 5 | \( 1 - 4.19T + 5T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 - 1.03T + 11T^{2} \) |
| 13 | \( 1 - 1.70T + 13T^{2} \) |
| 17 | \( 1 - 0.0209T + 17T^{2} \) |
| 19 | \( 1 - 4.05T + 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 + 5.01T + 29T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 + 6.49T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 3.83T + 47T^{2} \) |
| 53 | \( 1 + 7.11T + 53T^{2} \) |
| 59 | \( 1 - 5.35T + 59T^{2} \) |
| 61 | \( 1 + 2.81T + 61T^{2} \) |
| 67 | \( 1 - 2.91T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 2.84T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 3.26T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−8.212894095476913862308439819347, −7.24218209540098179707112710521, −6.47202332240197286290055088648, −5.86771010582574117123690532551, −5.52037107869870367211702872572, −4.59646560191973029556013585774, −3.32969890733372212132058700784, −2.78751356272249344037832731352, −1.91207541219205259113322386118, −0.938114656071443325068584044460,
0.938114656071443325068584044460, 1.91207541219205259113322386118, 2.78751356272249344037832731352, 3.32969890733372212132058700784, 4.59646560191973029556013585774, 5.52037107869870367211702872572, 5.86771010582574117123690532551, 6.47202332240197286290055088648, 7.24218209540098179707112710521, 8.212894095476913862308439819347
