L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s + 2·11-s − 6·13-s + 2·15-s + 2·17-s − 19-s + 21-s − 6·23-s − 25-s − 27-s − 8·29-s + 8·31-s − 2·33-s + 2·35-s − 10·37-s + 6·39-s + 8·41-s + 8·43-s − 2·45-s − 2·47-s + 49-s − 2·51-s − 8·53-s − 4·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.66·13-s + 0.516·15-s + 0.485·17-s − 0.229·19-s + 0.218·21-s − 1.25·23-s − 1/5·25-s − 0.192·27-s − 1.48·29-s + 1.43·31-s − 0.348·33-s + 0.338·35-s − 1.64·37-s + 0.960·39-s + 1.24·41-s + 1.21·43-s − 0.298·45-s − 0.291·47-s + 1/7·49-s − 0.280·51-s − 1.09·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5711191311\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5711191311\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.68232433615238804170965301329, −7.51424911854230365986047208393, −6.62504009228692607135522425819, −5.93577847671402433553300941243, −5.16355566525291829630601803053, −4.31338718100233797373331504302, −3.83197538552969534374276864475, −2.80963779812988033436297944151, −1.77927837757618721073394184152, −0.39207630584210326153724945313,
0.39207630584210326153724945313, 1.77927837757618721073394184152, 2.80963779812988033436297944151, 3.83197538552969534374276864475, 4.31338718100233797373331504302, 5.16355566525291829630601803053, 5.93577847671402433553300941243, 6.62504009228692607135522425819, 7.51424911854230365986047208393, 7.68232433615238804170965301329