L(s) = 1 | + 3-s + 2·7-s + 9-s + 2·11-s + 6·13-s − 2·17-s + 2·21-s − 4·23-s + 27-s + 8·31-s + 2·33-s + 2·37-s + 6·39-s + 2·41-s + 4·43-s − 8·47-s − 3·49-s − 2·51-s + 6·53-s + 10·59-s − 2·61-s + 2·63-s + 8·67-s − 4·69-s − 12·71-s + 4·73-s + 4·77-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.603·11-s + 1.66·13-s − 0.485·17-s + 0.436·21-s − 0.834·23-s + 0.192·27-s + 1.43·31-s + 0.348·33-s + 0.328·37-s + 0.960·39-s + 0.312·41-s + 0.609·43-s − 1.16·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s + 1.30·59-s − 0.256·61-s + 0.251·63-s + 0.977·67-s − 0.481·69-s − 1.42·71-s + 0.468·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.147331655\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.147331655\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.294621404044321153232941656353, −7.82916843904617298969738068421, −6.73848316801603166260924120780, −6.25629768791183300682565415383, −5.34862637188603959556918580152, −4.30363492128197640914363500647, −3.90195979683780024724235112717, −2.87323565715695326749103005451, −1.85419289855368654610516185561, −1.03766093095475251256503366567,
1.03766093095475251256503366567, 1.85419289855368654610516185561, 2.87323565715695326749103005451, 3.90195979683780024724235112717, 4.30363492128197640914363500647, 5.34862637188603959556918580152, 6.25629768791183300682565415383, 6.73848316801603166260924120780, 7.82916843904617298969738068421, 8.294621404044321153232941656353