L(s) = 1 | + 5-s + 4·7-s + 11-s + 6·17-s − 4·19-s + 4·23-s + 25-s − 2·29-s + 4·31-s + 4·35-s − 2·37-s + 8·41-s − 12·43-s + 9·49-s + 55-s + 4·59-s + 6·61-s + 6·67-s − 2·73-s + 4·77-s − 10·79-s + 6·83-s + 6·85-s + 6·89-s − 4·95-s + 6·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.301·11-s + 1.45·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s + 0.676·35-s − 0.328·37-s + 1.24·41-s − 1.82·43-s + 9/7·49-s + 0.134·55-s + 0.520·59-s + 0.768·61-s + 0.733·67-s − 0.234·73-s + 0.455·77-s − 1.12·79-s + 0.658·83-s + 0.650·85-s + 0.635·89-s − 0.410·95-s + 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.031943597\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.031943597\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.56558637304338, −12.03068702640340, −11.70967290992974, −11.20339836812560, −10.83504602369724, −10.33388073834900, −9.893590137622079, −9.455334042811070, −8.776591022810544, −8.484411422563388, −8.018535316893929, −7.604687601631135, −7.029781682232605, −6.573978538879062, −5.963590713122116, −5.444968937037789, −5.100437751099668, −4.594529495660213, −4.087775624804838, −3.460579248246720, −2.883851914566656, −2.193195334663458, −1.720252740192683, −1.184496812890090, −0.6228083069907842,
0.6228083069907842, 1.184496812890090, 1.720252740192683, 2.193195334663458, 2.883851914566656, 3.460579248246720, 4.087775624804838, 4.594529495660213, 5.100437751099668, 5.444968937037789, 5.963590713122116, 6.573978538879062, 7.029781682232605, 7.604687601631135, 8.018535316893929, 8.484411422563388, 8.776591022810544, 9.455334042811070, 9.893590137622079, 10.33388073834900, 10.83504602369724, 11.20339836812560, 11.70967290992974, 12.03068702640340, 12.56558637304338