L(s) = 1 | − 3-s + 9-s − 2·11-s − 4·13-s − 3·17-s + 5·19-s + 23-s − 5·25-s − 27-s − 7·29-s − 4·31-s + 2·33-s + 7·37-s + 4·39-s − 8·41-s − 2·43-s + 7·47-s − 7·49-s + 3·51-s − 14·53-s − 5·57-s − 15·59-s − 6·61-s + 67-s − 69-s + 9·73-s + 5·75-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.727·17-s + 1.14·19-s + 0.208·23-s − 25-s − 0.192·27-s − 1.29·29-s − 0.718·31-s + 0.348·33-s + 1.15·37-s + 0.640·39-s − 1.24·41-s − 0.304·43-s + 1.02·47-s − 49-s + 0.420·51-s − 1.92·53-s − 0.662·57-s − 1.95·59-s − 0.768·61-s + 0.122·67-s − 0.120·69-s + 1.05·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 67 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−9.759513125058147452977507413423, −9.274779884437189541788953179796, −7.85195808865438431644359870363, −7.36576282545777514322643063848, −6.25721615938415786811056368194, −5.32781026259762163268672672493, −4.59058515706647276941798356627, −3.25446015692750106863920902496, −1.91428236605633129322453542602, 0,
1.91428236605633129322453542602, 3.25446015692750106863920902496, 4.59058515706647276941798356627, 5.32781026259762163268672672493, 6.25721615938415786811056368194, 7.36576282545777514322643063848, 7.85195808865438431644359870363, 9.274779884437189541788953179796, 9.759513125058147452977507413423