| L(s) = 1 | + 2·11-s − 2·13-s + 2·19-s + 8·23-s − 2·29-s − 6·31-s + 8·37-s − 10·41-s + 12·47-s + 2·53-s − 2·61-s − 4·67-s + 14·71-s + 2·73-s − 4·79-s − 16·83-s − 6·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.603·11-s − 0.554·13-s + 0.458·19-s + 1.66·23-s − 0.371·29-s − 1.07·31-s + 1.31·37-s − 1.56·41-s + 1.75·47-s + 0.274·53-s − 0.256·61-s − 0.488·67-s + 1.66·71-s + 0.234·73-s − 0.450·79-s − 1.75·83-s − 0.635·89-s − 0.203·97-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
| \(L(1)\) |
\(\approx\) |
\(2.503995547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.503995547\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.13734941321027, −12.61159723353207, −12.41968262228080, −11.50545876625201, −11.49951991447737, −10.89715023238844, −10.29618914840445, −9.869613083976339, −9.277214661594928, −8.968540802267877, −8.542214299339497, −7.716175600088072, −7.416499026772469, −6.921266262901803, −6.451253983737974, −5.795662602600458, −5.227558775533397, −4.929084056765640, −4.106504042988343, −3.771192289574697, −2.990519246588010, −2.603244492228134, −1.784723931925931, −1.187608734849732, −0.4846259729308787,
0.4846259729308787, 1.187608734849732, 1.784723931925931, 2.603244492228134, 2.990519246588010, 3.771192289574697, 4.106504042988343, 4.929084056765640, 5.227558775533397, 5.795662602600458, 6.451253983737974, 6.921266262901803, 7.416499026772469, 7.716175600088072, 8.542214299339497, 8.968540802267877, 9.277214661594928, 9.869613083976339, 10.29618914840445, 10.89715023238844, 11.49951991447737, 11.50545876625201, 12.41968262228080, 12.61159723353207, 13.13734941321027
