| L(s) = 1 | + 2-s − 4-s − 3·8-s − 6·13-s − 16-s − 6·17-s − 19-s + 4·23-s − 6·26-s − 2·29-s + 8·31-s + 5·32-s − 6·34-s + 10·37-s − 38-s + 2·41-s + 4·43-s + 4·46-s + 12·47-s − 7·49-s + 6·52-s − 6·53-s − 2·58-s + 12·59-s − 2·61-s + 8·62-s + 7·64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1.66·13-s − 1/4·16-s − 1.45·17-s − 0.229·19-s + 0.834·23-s − 1.17·26-s − 0.371·29-s + 1.43·31-s + 0.883·32-s − 1.02·34-s + 1.64·37-s − 0.162·38-s + 0.312·41-s + 0.609·43-s + 0.589·46-s + 1.75·47-s − 49-s + 0.832·52-s − 0.824·53-s − 0.262·58-s + 1.56·59-s − 0.256·61-s + 1.01·62-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.553371989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.553371989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.413210533656607852401376939118, −7.61524143090288811055421546817, −6.79719281658243901645867852279, −6.11384960934632706858680338747, −5.21279738761597944732965739489, −4.57989855617964850469659731224, −4.12722114254805367027999161261, −2.87886881917875081784935438320, −2.34151253452678090701645815867, −0.60920853858641804094697968232,
0.60920853858641804094697968232, 2.34151253452678090701645815867, 2.87886881917875081784935438320, 4.12722114254805367027999161261, 4.57989855617964850469659731224, 5.21279738761597944732965739489, 6.11384960934632706858680338747, 6.79719281658243901645867852279, 7.61524143090288811055421546817, 8.413210533656607852401376939118
