L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 11-s − 14-s + 16-s − 4·17-s + 20-s + 22-s − 2·23-s + 25-s − 28-s − 6·29-s − 8·31-s + 32-s − 4·34-s − 35-s − 8·37-s + 40-s − 2·41-s − 4·43-s + 44-s − 2·46-s − 12·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s − 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.223·20-s + 0.213·22-s − 0.417·23-s + 1/5·25-s − 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.685·34-s − 0.169·35-s − 1.31·37-s + 0.158·40-s − 0.312·41-s − 0.609·43-s + 0.150·44-s − 0.294·46-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−7.44942006612015677551174412723, −6.59373002893840360060844518043, −6.36946009160999112604656838797, −5.32516283507146853281067324274, −4.92959856318581783556051933837, −3.80493541213602201479022356383, −3.41230318631145244014585090130, −2.23651483387559854843828793691, −1.65343364758712182889578261182, 0,
1.65343364758712182889578261182, 2.23651483387559854843828793691, 3.41230318631145244014585090130, 3.80493541213602201479022356383, 4.92959856318581783556051933837, 5.32516283507146853281067324274, 6.36946009160999112604656838797, 6.59373002893840360060844518043, 7.44942006612015677551174412723