| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s + 6·11-s + 13-s + 2·14-s + 16-s − 19-s + 6·22-s + 6·23-s + 26-s + 2·28-s − 3·29-s + 8·31-s + 32-s − 37-s − 38-s − 9·41-s + 8·43-s + 6·44-s + 6·46-s − 3·47-s − 3·49-s + 52-s + 3·53-s + 2·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 1.80·11-s + 0.277·13-s + 0.534·14-s + 1/4·16-s − 0.229·19-s + 1.27·22-s + 1.25·23-s + 0.196·26-s + 0.377·28-s − 0.557·29-s + 1.43·31-s + 0.176·32-s − 0.164·37-s − 0.162·38-s − 1.40·41-s + 1.21·43-s + 0.904·44-s + 0.884·46-s − 0.437·47-s − 3/7·49-s + 0.138·52-s + 0.412·53-s + 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.315260887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.315260887\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.042621090760022367342815226829, −7.23919104834280773181693704759, −6.54207741440573394352993000704, −6.06219550173066916742324036953, −5.03838021306888140188406543366, −4.50158289562940385042348016331, −3.75592039875061832429293526542, −2.97904699739707532993955643879, −1.80030567957567006808966075770, −1.10369706929804914245289066162,
1.10369706929804914245289066162, 1.80030567957567006808966075770, 2.97904699739707532993955643879, 3.75592039875061832429293526542, 4.50158289562940385042348016331, 5.03838021306888140188406543366, 6.06219550173066916742324036953, 6.54207741440573394352993000704, 7.23919104834280773181693704759, 8.042621090760022367342815226829
