L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 4·7-s + 8-s + 9-s + 12-s − 4·13-s − 4·14-s + 16-s + 6·17-s + 18-s + 19-s − 4·21-s − 6·23-s + 24-s − 5·25-s − 4·26-s + 27-s − 4·28-s + 6·29-s + 2·31-s + 32-s + 6·34-s + 36-s − 4·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s − 1.10·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.229·19-s − 0.872·21-s − 1.25·23-s + 0.204·24-s − 25-s − 0.784·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s + 0.359·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.574435356\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.574435356\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.70443556114571377556292054916, −12.53921727142600454148846778560, −12.03323217483777155326327378475, −10.14217150491024881359990946724, −9.659160529040683538070775225454, −7.973335490421157373200340268575, −6.84732233829542762166868447522, −5.61719283222003359694793802526, −3.89674503164637365373404147869, −2.72251484544452322835217288408,
2.72251484544452322835217288408, 3.89674503164637365373404147869, 5.61719283222003359694793802526, 6.84732233829542762166868447522, 7.973335490421157373200340268575, 9.659160529040683538070775225454, 10.14217150491024881359990946724, 12.03323217483777155326327378475, 12.53921727142600454148846778560, 13.70443556114571377556292054916