| L(s) = 1 | − 4·11-s + 4·13-s + 4·19-s − 2·29-s + 8·31-s + 6·37-s + 4·43-s − 8·47-s − 10·53-s − 4·59-s + 4·61-s + 4·67-s + 8·71-s − 16·73-s + 8·79-s − 12·83-s + 8·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 1.10·13-s + 0.917·19-s − 0.371·29-s + 1.43·31-s + 0.986·37-s + 0.609·43-s − 1.16·47-s − 1.37·53-s − 0.520·59-s + 0.512·61-s + 0.488·67-s + 0.949·71-s − 1.87·73-s + 0.900·79-s − 1.31·83-s + 0.847·89-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.375968748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.375968748\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−13.12792837822637, −12.93806864021744, −12.21535982011991, −11.70966283497430, −11.27942957601091, −10.85449774348252, −10.36644292262322, −9.869870120793701, −9.432075312777978, −8.889010552160656, −8.267782189918356, −7.865783805923658, −7.630086160271014, −6.767675629739228, −6.390804019762973, −5.798929533595631, −5.371966222243507, −4.773902843085592, −4.299595933954702, −3.583475972928436, −3.023097507537502, −2.635918434868129, −1.807602293170575, −1.153143068454814, −0.4779893551997866,
0.4779893551997866, 1.153143068454814, 1.807602293170575, 2.635918434868129, 3.023097507537502, 3.583475972928436, 4.299595933954702, 4.773902843085592, 5.371966222243507, 5.798929533595631, 6.390804019762973, 6.767675629739228, 7.630086160271014, 7.865783805923658, 8.267782189918356, 8.889010552160656, 9.432075312777978, 9.869870120793701, 10.36644292262322, 10.85449774348252, 11.27942957601091, 11.70966283497430, 12.21535982011991, 12.93806864021744, 13.12792837822637
