| L(s) = 1 | + 2·5-s − 4·11-s − 6·13-s + 4·17-s + 6·19-s − 4·23-s − 25-s − 6·29-s + 4·31-s + 6·37-s − 4·41-s + 12·43-s − 12·47-s + 6·53-s − 8·55-s − 6·59-s + 6·61-s − 12·65-s + 12·67-s + 8·71-s − 6·83-s + 8·85-s + 16·89-s + 12·95-s − 12·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.20·11-s − 1.66·13-s + 0.970·17-s + 1.37·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.986·37-s − 0.624·41-s + 1.82·43-s − 1.75·47-s + 0.824·53-s − 1.07·55-s − 0.781·59-s + 0.768·61-s − 1.48·65-s + 1.46·67-s + 0.949·71-s − 0.658·83-s + 0.867·85-s + 1.69·89-s + 1.23·95-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−15.52404195282142, −14.86361004639690, −14.27159449574615, −14.03767130633125, −13.30152691879750, −12.89908260900750, −12.31187467101445, −11.78972585784141, −11.24634478706192, −10.38586591493197, −9.989830464319888, −9.628002670252124, −9.253858215026559, −8.076796642283467, −7.812962258815990, −7.349142163148044, −6.553189465683261, −5.779491183218510, −5.326954135036494, −5.010984132039218, −4.067496984506897, −3.207870175588840, −2.517500753178658, −2.093342995697484, −1.051836562559962, 0,
1.051836562559962, 2.093342995697484, 2.517500753178658, 3.207870175588840, 4.067496984506897, 5.010984132039218, 5.326954135036494, 5.779491183218510, 6.553189465683261, 7.349142163148044, 7.812962258815990, 8.076796642283467, 9.253858215026559, 9.628002670252124, 9.989830464319888, 10.38586591493197, 11.24634478706192, 11.78972585784141, 12.31187467101445, 12.89908260900750, 13.30152691879750, 14.03767130633125, 14.27159449574615, 14.86361004639690, 15.52404195282142
