| L(s) = 1 | + 3.33·5-s + 2.77·7-s − 2.02·11-s − 3.10·13-s + 7.91·17-s + 2.17·19-s + 4.75·23-s + 6.10·25-s − 3.85·29-s − 10.6·31-s + 9.26·35-s + 6.58·37-s + 41-s + 4.80·43-s − 4.03·47-s + 0.719·49-s + 1.94·53-s − 6.76·55-s − 4.75·59-s + 2.89·61-s − 10.3·65-s + 5.97·67-s − 14.4·71-s − 9.24·73-s − 5.63·77-s − 0.320·79-s + 3.69·83-s + ⋯ |
| L(s) = 1 | + 1.49·5-s + 1.05·7-s − 0.611·11-s − 0.862·13-s + 1.91·17-s + 0.500·19-s + 0.990·23-s + 1.22·25-s − 0.716·29-s − 1.91·31-s + 1.56·35-s + 1.08·37-s + 0.156·41-s + 0.733·43-s − 0.589·47-s + 0.102·49-s + 0.266·53-s − 0.911·55-s − 0.618·59-s + 0.370·61-s − 1.28·65-s + 0.729·67-s − 1.70·71-s − 1.08·73-s − 0.642·77-s − 0.0360·79-s + 0.405·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.503254862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.503254862\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 5 | \( 1 - 3.33T + 5T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 11 | \( 1 + 2.02T + 11T^{2} \) |
| 13 | \( 1 + 3.10T + 13T^{2} \) |
| 17 | \( 1 - 7.91T + 17T^{2} \) |
| 19 | \( 1 - 2.17T + 19T^{2} \) |
| 23 | \( 1 - 4.75T + 23T^{2} \) |
| 29 | \( 1 + 3.85T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 - 6.58T + 37T^{2} \) |
| 43 | \( 1 - 4.80T + 43T^{2} \) |
| 47 | \( 1 + 4.03T + 47T^{2} \) |
| 53 | \( 1 - 1.94T + 53T^{2} \) |
| 59 | \( 1 + 4.75T + 59T^{2} \) |
| 61 | \( 1 - 2.89T + 61T^{2} \) |
| 67 | \( 1 - 5.97T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 9.24T + 73T^{2} \) |
| 79 | \( 1 + 0.320T + 79T^{2} \) |
| 83 | \( 1 - 3.69T + 83T^{2} \) |
| 89 | \( 1 + 8.96T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−9.601152593009975931339769103575, −8.864467165348746399329867489012, −7.67270611579207164166519903527, −7.35274369508745377605573694445, −5.89870684372240246171444060112, −5.42856810522578902295457436340, −4.78740242364405846165531092155, −3.25170939291966579694240912121, −2.21463319068287691673727535708, −1.28281547087359610777505235269,
1.28281547087359610777505235269, 2.21463319068287691673727535708, 3.25170939291966579694240912121, 4.78740242364405846165531092155, 5.42856810522578902295457436340, 5.89870684372240246171444060112, 7.35274369508745377605573694445, 7.67270611579207164166519903527, 8.864467165348746399329867489012, 9.601152593009975931339769103575
