| L(s) = 1 | + 7-s + 4·11-s − 2·13-s − 8·17-s + 6·19-s − 4·23-s − 2·29-s − 4·31-s + 8·37-s + 10·41-s − 10·43-s + 10·47-s + 49-s − 2·53-s − 4·59-s + 4·61-s − 10·67-s + 6·71-s − 6·73-s + 4·77-s + 8·79-s + 8·83-s − 6·89-s − 2·91-s + 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s − 0.554·13-s − 1.94·17-s + 1.37·19-s − 0.834·23-s − 0.371·29-s − 0.718·31-s + 1.31·37-s + 1.56·41-s − 1.52·43-s + 1.45·47-s + 1/7·49-s − 0.274·53-s − 0.520·59-s + 0.512·61-s − 1.22·67-s + 0.712·71-s − 0.702·73-s + 0.455·77-s + 0.900·79-s + 0.878·83-s − 0.635·89-s − 0.209·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.213074204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213074204\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.41532274580943, −14.10151343958683, −13.51128478036276, −13.05925420028896, −12.39553318931571, −11.85033498516107, −11.46394703776911, −11.05059954093052, −10.42747485192309, −9.679211398918207, −9.268745036999719, −8.938983703533533, −8.226677425429881, −7.475839901835198, −7.273429546681330, −6.401978481576178, −6.104734261333688, −5.300849359901634, −4.673981824200608, −4.146229482844158, −3.645729217434533, −2.709860550630678, −2.115121807593142, −1.421077973467734, −0.5300683338814438,
0.5300683338814438, 1.421077973467734, 2.115121807593142, 2.709860550630678, 3.645729217434533, 4.146229482844158, 4.673981824200608, 5.300849359901634, 6.104734261333688, 6.401978481576178, 7.273429546681330, 7.475839901835198, 8.226677425429881, 8.938983703533533, 9.268745036999719, 9.679211398918207, 10.42747485192309, 11.05059954093052, 11.46394703776911, 11.85033498516107, 12.39553318931571, 13.05925420028896, 13.51128478036276, 14.10151343958683, 14.41532274580943
