| L(s) = 1 | + 2·13-s − 2·17-s − 6·19-s + 6·23-s + 2·29-s − 2·31-s − 2·37-s + 2·41-s − 8·43-s − 8·67-s − 10·73-s + 8·79-s + 12·83-s − 6·89-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.554·13-s − 0.485·17-s − 1.37·19-s + 1.25·23-s + 0.371·29-s − 0.359·31-s − 0.328·37-s + 0.312·41-s − 1.21·43-s − 0.977·67-s − 1.17·73-s + 0.900·79-s + 1.31·83-s − 0.635·89-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.40238081559289, −12.99364892074656, −12.54175808224974, −12.01848555719956, −11.48724336469956, −10.98016230529110, −10.65640731373524, −10.23413992426403, −9.551293240620746, −9.060502627699497, −8.605375727879934, −8.333330415655422, −7.597033713696942, −7.112116541186843, −6.517276303633459, −6.284557941049801, −5.562314721697537, −5.014785168203252, −4.486795888965152, −4.014900496242945, −3.323548608983093, −2.857052136325260, −2.097379039487941, −1.597834096114033, −0.7950723189749931, 0,
0.7950723189749931, 1.597834096114033, 2.097379039487941, 2.857052136325260, 3.323548608983093, 4.014900496242945, 4.486795888965152, 5.014785168203252, 5.562314721697537, 6.284557941049801, 6.517276303633459, 7.112116541186843, 7.597033713696942, 8.333330415655422, 8.605375727879934, 9.060502627699497, 9.551293240620746, 10.23413992426403, 10.65640731373524, 10.98016230529110, 11.48724336469956, 12.01848555719956, 12.54175808224974, 12.99364892074656, 13.40238081559289
