L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 5.16·11-s − 4.16·13-s + 14-s + 16-s − 5·17-s − 7.16·19-s + 5.16·22-s − 6.16·23-s − 4.16·26-s + 28-s + 4.16·29-s + 10.4·31-s + 32-s − 5·34-s − 2.83·37-s − 7.16·38-s − 10.3·41-s − 7·43-s + 5.16·44-s − 6.16·46-s + 2·47-s + 49-s − 4.16·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.377·7-s + 0.353·8-s + 1.55·11-s − 1.15·13-s + 0.267·14-s + 0.250·16-s − 1.21·17-s − 1.64·19-s + 1.10·22-s − 1.28·23-s − 0.816·26-s + 0.188·28-s + 0.772·29-s + 1.88·31-s + 0.176·32-s − 0.857·34-s − 0.466·37-s − 1.16·38-s − 1.61·41-s − 1.06·43-s + 0.778·44-s − 0.908·46-s + 0.291·47-s + 0.142·49-s − 0.577·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 5.16T + 11T^{2} \) |
| 13 | \( 1 + 4.16T + 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 + 7.16T + 19T^{2} \) |
| 23 | \( 1 + 6.16T + 23T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 2.83T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 7T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 6.48T + 53T^{2} \) |
| 59 | \( 1 + 7T + 59T^{2} \) |
| 61 | \( 1 - 4.32T + 61T^{2} \) |
| 67 | \( 1 + 2.67T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 4.32T + 83T^{2} \) |
| 89 | \( 1 + 2.67T + 89T^{2} \) |
| 97 | \( 1 - 5.16T + 97T^{2} \) |
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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
−7.08323139593088911593461231996, −6.45435504146751086895342964227, −6.28827778407599451143684437855, −5.08474127521247381947592990987, −4.39827583339048642594067271675, −4.17907284765987261173876221227, −3.06405774165612595731684759890, −2.18256989492587604085649459610, −1.55362884393722454543414939829, 0,
1.55362884393722454543414939829, 2.18256989492587604085649459610, 3.06405774165612595731684759890, 4.17907284765987261173876221227, 4.39827583339048642594067271675, 5.08474127521247381947592990987, 6.28827778407599451143684437855, 6.45435504146751086895342964227, 7.08323139593088911593461231996