| L(s) = 1 | − 3-s − 7-s + 9-s − 2·11-s + 6·13-s + 4·17-s + 4·19-s + 21-s + 2·23-s − 27-s − 2·29-s + 2·33-s − 2·37-s − 6·39-s − 4·43-s + 12·47-s + 49-s − 4·51-s + 6·53-s − 4·57-s + 8·59-s + 6·61-s − 63-s − 8·67-s − 2·69-s − 14·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.66·13-s + 0.970·17-s + 0.917·19-s + 0.218·21-s + 0.417·23-s − 0.192·27-s − 0.371·29-s + 0.348·33-s − 0.328·37-s − 0.960·39-s − 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.560·51-s + 0.824·53-s − 0.529·57-s + 1.04·59-s + 0.768·61-s − 0.125·63-s − 0.977·67-s − 0.240·69-s − 1.66·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.758261932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.758261932\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.62008507230978167192824949618, −7.16560439575924885821026398481, −6.29566344437301545711428632951, −5.65008184920859047762434349872, −5.29678943849831044830329432095, −4.19016081135333242837427210842, −3.52504133685650821098843731061, −2.79508532036365372719773263070, −1.51804244031740427581273441383, −0.72068481015456117778800650454,
0.72068481015456117778800650454, 1.51804244031740427581273441383, 2.79508532036365372719773263070, 3.52504133685650821098843731061, 4.19016081135333242837427210842, 5.29678943849831044830329432095, 5.65008184920859047762434349872, 6.29566344437301545711428632951, 7.16560439575924885821026398481, 7.62008507230978167192824949618
