| L(s) = 1 | − 3-s + 7-s + 9-s + 6·11-s − 2·13-s + 4·17-s + 6·19-s − 21-s − 27-s − 2·29-s + 10·31-s − 6·33-s − 4·37-s + 2·39-s + 2·41-s + 4·43-s + 49-s − 4·51-s + 6·53-s − 6·57-s + 8·59-s − 2·61-s + 63-s + 16·67-s − 10·71-s − 6·73-s + 6·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 0.970·17-s + 1.37·19-s − 0.218·21-s − 0.192·27-s − 0.371·29-s + 1.79·31-s − 1.04·33-s − 0.657·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s + 1/7·49-s − 0.560·51-s + 0.824·53-s − 0.794·57-s + 1.04·59-s − 0.256·61-s + 0.125·63-s + 1.95·67-s − 1.18·71-s − 0.702·73-s + 0.683·77-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
| \(L(1)\) |
\(\approx\) |
\(2.322361287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.322361287\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.63642208399059987664156552537, −7.08348656460081845011295837696, −6.43826823199024400071515549204, −5.68036975366865001334852324611, −5.10097823060024598246390702687, −4.26107229251121088868386881866, −3.64436103322453345015934958636, −2.67374112901219525123470788727, −1.44940318569212649034179954751, −0.879874916978760848533151865289,
0.879874916978760848533151865289, 1.44940318569212649034179954751, 2.67374112901219525123470788727, 3.64436103322453345015934958636, 4.26107229251121088868386881866, 5.10097823060024598246390702687, 5.68036975366865001334852324611, 6.43826823199024400071515549204, 7.08348656460081845011295837696, 7.63642208399059987664156552537
