| L(s) = 1 | − i·3-s − 4i·7-s − 9-s − 2i·13-s + 6i·17-s − 2·19-s − 4·21-s + i·23-s + i·27-s − 6·29-s − 4·31-s + 8i·37-s − 2·39-s + 6·41-s − 8i·43-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.51i·7-s − 0.333·9-s − 0.554i·13-s + 1.45i·17-s − 0.458·19-s − 0.872·21-s + 0.208i·23-s + 0.192i·27-s − 1.11·29-s − 0.718·31-s + 1.31i·37-s − 0.320·39-s + 0.937·41-s − 1.21i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7550602305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7550602305\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−7.84325177004026748885720684217, −7.51099364120308993359537574774, −6.74370874392025267337939275358, −6.09077571627175807068386095491, −5.38202517201003626289371329611, −4.26983774652200597141317345311, −3.84621089162059485664098388963, −2.91972584270743481853768013241, −1.75652702117879535468281759557, −1.02316651823771868751128250463,
0.19586715114141976357915167741, 1.92236265836866100869575257903, 2.53416881449374513532073560176, 3.40207338858494753362436711250, 4.28975302144482525278030970395, 5.11172777273964489799533236863, 5.58304154348854151435419205529, 6.30772265812298919956730078128, 7.14572339449530757266997410522, 7.894618990271258838482362332938
