| L(s) = 1 | − 5·7-s − 5·13-s − 19-s + 7·31-s + 10·37-s + 5·43-s + 18·49-s + 13·61-s + 5·67-s − 10·73-s + 4·79-s + 25·91-s + 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.88·7-s − 1.38·13-s − 0.229·19-s + 1.25·31-s + 1.64·37-s + 0.762·43-s + 18/7·49-s + 1.66·61-s + 0.610·67-s − 1.17·73-s + 0.450·79-s + 2.62·91-s + 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−16.40983526324509, −15.86426555753760, −15.36081012124829, −14.75948538989718, −14.17171412371318, −13.48462051146373, −12.90200712879736, −12.63030232220654, −11.96759659766280, −11.43272244985376, −10.47865367269879, −10.03719907943383, −9.586527745847654, −9.139774415798464, −8.302845551815038, −7.552149934304362, −6.986297696879915, −6.396216413826226, −5.901833722839695, −5.063818959964122, −4.281767880873783, −3.621239578514575, −2.670628960081663, −2.479473089699288, −0.9077754518684141, 0,
0.9077754518684141, 2.479473089699288, 2.670628960081663, 3.621239578514575, 4.281767880873783, 5.063818959964122, 5.901833722839695, 6.396216413826226, 6.986297696879915, 7.552149934304362, 8.302845551815038, 9.139774415798464, 9.586527745847654, 10.03719907943383, 10.47865367269879, 11.43272244985376, 11.96759659766280, 12.63030232220654, 12.90200712879736, 13.48462051146373, 14.17171412371318, 14.75948538989718, 15.36081012124829, 15.86426555753760, 16.40983526324509
