L(s) = 1 | + 2-s + 4-s + 5-s − 4·7-s + 8-s + 10-s − 4·11-s + 2·13-s − 4·14-s + 16-s + 2·17-s + 8·19-s + 20-s − 4·22-s + 25-s + 2·26-s − 4·28-s + 6·29-s − 8·31-s + 32-s + 2·34-s − 4·35-s − 37-s + 8·38-s + 40-s + 6·41-s + 4·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s + 0.353·8-s + 0.316·10-s − 1.20·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.485·17-s + 1.83·19-s + 0.223·20-s − 0.852·22-s + 1/5·25-s + 0.392·26-s − 0.755·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.342·34-s − 0.676·35-s − 0.164·37-s + 1.29·38-s + 0.158·40-s + 0.937·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.682298596\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.682298596\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.720628321985890445851332037026, −7.51863494409539931971416508018, −7.19362759074251700470132931200, −6.05952525756235498651268414822, −5.73540830164809785044014825167, −4.95211480680359505928568260645, −3.74035267532993164482938008679, −3.11567405464547249517948337351, −2.40998484381432637993100803339, −0.875384951555919833294195341928,
0.875384951555919833294195341928, 2.40998484381432637993100803339, 3.11567405464547249517948337351, 3.74035267532993164482938008679, 4.95211480680359505928568260645, 5.73540830164809785044014825167, 6.05952525756235498651268414822, 7.19362759074251700470132931200, 7.51863494409539931971416508018, 8.720628321985890445851332037026