L(s) = 1 | − 5-s − 7-s + 3·11-s − 13-s + 3·17-s − 2·19-s − 6·23-s + 25-s + 9·29-s − 8·31-s + 35-s − 10·37-s − 2·43-s − 3·47-s + 49-s − 3·55-s + 12·59-s + 8·61-s + 65-s − 8·67-s + 14·73-s − 3·77-s − 5·79-s − 12·83-s − 3·85-s − 12·89-s + 91-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.904·11-s − 0.277·13-s + 0.727·17-s − 0.458·19-s − 1.25·23-s + 1/5·25-s + 1.67·29-s − 1.43·31-s + 0.169·35-s − 1.64·37-s − 0.304·43-s − 0.437·47-s + 1/7·49-s − 0.404·55-s + 1.56·59-s + 1.02·61-s + 0.124·65-s − 0.977·67-s + 1.63·73-s − 0.341·77-s − 0.562·79-s − 1.31·83-s − 0.325·85-s − 1.27·89-s + 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−7.939416143880326623174393786183, −7.04505267077565989088641618725, −6.58121324629163679591857734879, −5.72087922152288564809689768968, −4.92814931116241942042627067745, −3.94857875855529863289243822571, −3.51448804427200098656523780783, −2.40184661461123738410720377579, −1.32849554317303336039309185031, 0,
1.32849554317303336039309185031, 2.40184661461123738410720377579, 3.51448804427200098656523780783, 3.94857875855529863289243822571, 4.92814931116241942042627067745, 5.72087922152288564809689768968, 6.58121324629163679591857734879, 7.04505267077565989088641618725, 7.939416143880326623174393786183