| L(s) = 1 | − 2-s + 4-s − 8-s + 6·11-s + 13-s + 16-s + 6·19-s − 6·22-s − 6·23-s − 26-s − 2·29-s + 4·31-s − 32-s + 10·37-s − 6·38-s + 6·41-s + 8·43-s + 6·44-s + 6·46-s − 8·47-s − 7·49-s + 52-s + 6·53-s + 2·58-s − 10·59-s − 6·61-s − 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.80·11-s + 0.277·13-s + 1/4·16-s + 1.37·19-s − 1.27·22-s − 1.25·23-s − 0.196·26-s − 0.371·29-s + 0.718·31-s − 0.176·32-s + 1.64·37-s − 0.973·38-s + 0.937·41-s + 1.21·43-s + 0.904·44-s + 0.884·46-s − 1.16·47-s − 49-s + 0.138·52-s + 0.824·53-s + 0.262·58-s − 1.30·59-s − 0.768·61-s − 0.508·62-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
| \(L(1)\) |
\(\approx\) |
\(1.745457698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.745457698\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.914860535659067241209585074300, −7.70302577369861880861944948021, −6.49453266589921559642006169883, −6.34812805336864913009556585870, −5.37825277026734998627498746353, −4.26975023624364733927794525996, −3.67493208538793214417483424997, −2.68610735279515919365131723282, −1.58943084195169391148044960545, −0.843958229069094735950754984481,
0.843958229069094735950754984481, 1.58943084195169391148044960545, 2.68610735279515919365131723282, 3.67493208538793214417483424997, 4.26975023624364733927794525996, 5.37825277026734998627498746353, 6.34812805336864913009556585870, 6.49453266589921559642006169883, 7.70302577369861880861944948021, 7.914860535659067241209585074300
