| L(s) = 1 | − 3-s − 7-s + 9-s + 4·11-s − 2·13-s + 6·17-s + 4·19-s + 21-s − 27-s + 2·29-s − 4·33-s + 6·37-s + 2·39-s + 2·41-s + 4·43-s + 49-s − 6·51-s + 6·53-s − 4·57-s + 12·59-s + 2·61-s − 63-s − 4·67-s + 6·73-s − 4·77-s + 16·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.218·21-s − 0.192·27-s + 0.371·29-s − 0.696·33-s + 0.986·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s + 1/7·49-s − 0.840·51-s + 0.824·53-s − 0.529·57-s + 1.56·59-s + 0.256·61-s − 0.125·63-s − 0.488·67-s + 0.702·73-s − 0.455·77-s + 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.417232116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.417232116\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.80430125866463, −14.65155755939188, −13.84387391288792, −13.59096735799922, −12.57307332757042, −12.40829950728620, −11.85216658348249, −11.40877075732401, −10.81154087425392, −10.02263817222484, −9.729496135738963, −9.297108511986945, −8.526768756806815, −7.821558088898063, −7.290231458818911, −6.771267127041178, −6.135923767757036, −5.586701123300880, −5.075183328190938, −4.251207184832909, −3.710244808182174, −3.041067267626762, −2.208653685269620, −1.160547307808244, −0.7247743354766993,
0.7247743354766993, 1.160547307808244, 2.208653685269620, 3.041067267626762, 3.710244808182174, 4.251207184832909, 5.075183328190938, 5.586701123300880, 6.135923767757036, 6.771267127041178, 7.290231458818911, 7.821558088898063, 8.526768756806815, 9.297108511986945, 9.729496135738963, 10.02263817222484, 10.81154087425392, 11.40877075732401, 11.85216658348249, 12.40829950728620, 12.57307332757042, 13.59096735799922, 13.84387391288792, 14.65155755939188, 14.80430125866463
