| L(s) = 1 | − 3-s − 5-s + 9-s + 15-s − 2·23-s + 25-s − 27-s + 2·29-s + 2·43-s − 45-s + 2·47-s + 49-s + 2·67-s + 2·69-s − 75-s + 81-s − 2·87-s − 2·101-s + 2·115-s + ⋯ |
| L(s) = 1 | − 3-s − 5-s + 9-s + 15-s − 2·23-s + 25-s − 27-s + 2·29-s + 2·43-s − 45-s + 2·47-s + 49-s + 2·67-s + 2·69-s − 75-s + 81-s − 2·87-s − 2·101-s + 2·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6569225496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6569225496\) |
| \(L(1)\) |
|
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 | \( 1 + T \) |
| good | 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 + T )^{2} \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.497038052068493434370284423254, −8.458301559314259211394116563717, −7.76135505342052605239548751680, −7.01447106605548671135866618651, −6.20668721293848894855487822980, −5.42071781825417558844578767926, −4.35187752719025936406305780436, −3.93470869018669441692022106635, −2.47446249427073202431405980879, −0.853544766397543477419339193187,
0.853544766397543477419339193187, 2.47446249427073202431405980879, 3.93470869018669441692022106635, 4.35187752719025936406305780436, 5.42071781825417558844578767926, 6.20668721293848894855487822980, 7.01447106605548671135866618651, 7.76135505342052605239548751680, 8.458301559314259211394116563717, 9.497038052068493434370284423254
