| L(s) = 1 | + 5.29·11-s − 5.29·23-s − 5·25-s − 10.5·29-s + 6·37-s − 12·43-s − 10.5·53-s − 4·67-s + 5.29·71-s − 8·79-s + 5.29·107-s − 18·109-s + 21.1·113-s + ⋯ |
| L(s) = 1 | + 1.59·11-s − 1.10·23-s − 25-s − 1.96·29-s + 0.986·37-s − 1.82·43-s − 1.45·53-s − 0.488·67-s + 0.627·71-s − 0.900·79-s + 0.511·107-s − 1.72·109-s + 1.99·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 5.29T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 5.29T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.63190617821405116546613461469, −6.84038130678808658740600265227, −6.17905657039650798292403368024, −5.64805452130056474265119441868, −4.60459101646859554227310891498, −3.91000853454298680299180833282, −3.36837569499946187345998149396, −2.07675938912727810074456136871, −1.43225651496463892881785693542, 0,
1.43225651496463892881785693542, 2.07675938912727810074456136871, 3.36837569499946187345998149396, 3.91000853454298680299180833282, 4.60459101646859554227310891498, 5.64805452130056474265119441868, 6.17905657039650798292403368024, 6.84038130678808658740600265227, 7.63190617821405116546613461469
