| L(s) = 1 | − 2·3-s + 7-s + 9-s + 11-s + 4·13-s + 4·19-s − 2·21-s + 4·27-s − 6·29-s + 10·31-s − 2·33-s − 2·37-s − 8·39-s − 12·41-s − 4·43-s + 6·47-s + 49-s + 6·53-s − 8·57-s + 6·59-s − 4·61-s + 63-s − 4·67-s − 12·71-s + 4·73-s + 77-s − 8·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.917·19-s − 0.436·21-s + 0.769·27-s − 1.11·29-s + 1.79·31-s − 0.348·33-s − 0.328·37-s − 1.28·39-s − 1.87·41-s − 0.609·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s − 1.05·57-s + 0.781·59-s − 0.512·61-s + 0.125·63-s − 0.488·67-s − 1.42·71-s + 0.468·73-s + 0.113·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.657617244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.657617244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−15.28447304708807, −14.57395104195307, −13.96674633747016, −13.45021067601467, −13.10048081496047, −12.11785764656676, −11.81182479507465, −11.55009899279523, −10.90737243452518, −10.32135603135949, −9.998634821305627, −9.011652290591431, −8.677874383298734, −8.006831436968514, −7.291683138232883, −6.723734772611749, −6.080007592622997, −5.732459365736906, −5.004538834516158, −4.580935216073206, −3.661356023122089, −3.163513030553258, −2.062778099631307, −1.247188800808953, −0.5952614736004134,
0.5952614736004134, 1.247188800808953, 2.062778099631307, 3.163513030553258, 3.661356023122089, 4.580935216073206, 5.004538834516158, 5.732459365736906, 6.080007592622997, 6.723734772611749, 7.291683138232883, 8.006831436968514, 8.677874383298734, 9.011652290591431, 9.998634821305627, 10.32135603135949, 10.90737243452518, 11.55009899279523, 11.81182479507465, 12.11785764656676, 13.10048081496047, 13.45021067601467, 13.96674633747016, 14.57395104195307, 15.28447304708807
