| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 5·11-s + 5·13-s + 15-s + 17-s − 2·19-s + 21-s + 2·23-s − 4·25-s + 27-s − 2·29-s − 8·31-s + 5·33-s + 35-s − 7·37-s + 5·39-s − 12·41-s + 5·43-s + 45-s + 12·47-s + 49-s + 51-s + 53-s + 5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.50·11-s + 1.38·13-s + 0.258·15-s + 0.242·17-s − 0.458·19-s + 0.218·21-s + 0.417·23-s − 4/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.870·33-s + 0.169·35-s − 1.15·37-s + 0.800·39-s − 1.87·41-s + 0.762·43-s + 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.140·51-s + 0.137·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.256089193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.256089193\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−15.50865811042794, −14.81619585010163, −14.33695790654175, −13.97577088174287, −13.39912087588392, −12.94186678793483, −12.22458291821833, −11.61748801555732, −11.19250779466893, −10.47158946537552, −10.00351169805091, −9.164847808072507, −8.837203568156261, −8.514322999294889, −7.599893965105097, −7.035211203810744, −6.446830502130342, −5.795493219195367, −5.255995451670441, −4.217154640648233, −3.768262465377708, −3.301850208999797, −2.049812631297612, −1.707148725523155, −0.8448935214232701,
0.8448935214232701, 1.707148725523155, 2.049812631297612, 3.301850208999797, 3.768262465377708, 4.217154640648233, 5.255995451670441, 5.795493219195367, 6.446830502130342, 7.035211203810744, 7.599893965105097, 8.514322999294889, 8.837203568156261, 9.164847808072507, 10.00351169805091, 10.47158946537552, 11.19250779466893, 11.61748801555732, 12.22458291821833, 12.94186678793483, 13.39912087588392, 13.97577088174287, 14.33695790654175, 14.81619585010163, 15.50865811042794
