| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 2·13-s − 14-s + 16-s + 2·19-s − 2·26-s − 28-s + 6·29-s + 8·31-s + 32-s + 4·37-s + 2·38-s + 6·41-s − 2·43-s + 6·47-s + 49-s − 2·52-s − 6·53-s − 56-s + 6·58-s + 12·59-s + 8·61-s + 8·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.458·19-s − 0.392·26-s − 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.657·37-s + 0.324·38-s + 0.937·41-s − 0.304·43-s + 0.875·47-s + 1/7·49-s − 0.277·52-s − 0.824·53-s − 0.133·56-s + 0.787·58-s + 1.56·59-s + 1.02·61-s + 1.01·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.854621188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.854621188\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.583295435803264526433628743831, −7.83484788492072391147649423168, −7.03243190705667838716624797088, −6.38185295352556876478721788069, −5.60655308738568935898973053621, −4.77365969038311726734736492284, −4.07381421480739036927784063240, −3.03657073871473273126945511382, −2.37713756717188893339656403534, −0.932612211263586597747614963660,
0.932612211263586597747614963660, 2.37713756717188893339656403534, 3.03657073871473273126945511382, 4.07381421480739036927784063240, 4.77365969038311726734736492284, 5.60655308738568935898973053621, 6.38185295352556876478721788069, 7.03243190705667838716624797088, 7.83484788492072391147649423168, 8.583295435803264526433628743831
