L(s) = 1 | + 3-s + 5-s − 2·9-s − 5·11-s + 2·13-s + 15-s + 3·17-s + 3·19-s − 23-s − 4·25-s − 5·27-s − 10·29-s + 11·31-s − 5·33-s + 7·37-s + 2·39-s + 41-s + 8·43-s − 2·45-s + 7·47-s + 3·51-s − 11·53-s − 5·55-s + 3·57-s + 7·59-s + 61-s + 2·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s − 1.50·11-s + 0.554·13-s + 0.258·15-s + 0.727·17-s + 0.688·19-s − 0.208·23-s − 4/5·25-s − 0.962·27-s − 1.85·29-s + 1.97·31-s − 0.870·33-s + 1.15·37-s + 0.320·39-s + 0.156·41-s + 1.21·43-s − 0.298·45-s + 1.02·47-s + 0.420·51-s − 1.51·53-s − 0.674·55-s + 0.397·57-s + 0.911·59-s + 0.128·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.283022571\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.283022571\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−7.77593179438390054962450948682, −7.53016255729802160081509384691, −6.22827256931015434938578234610, −5.74240001366221175143832145237, −5.23035060020067689268665763590, −4.19799837391576246663386741899, −3.32322767400617550083781111066, −2.67916537566897855360992727818, −2.00412240876726749654446199912, −0.70520324392681296805677422701,
0.70520324392681296805677422701, 2.00412240876726749654446199912, 2.67916537566897855360992727818, 3.32322767400617550083781111066, 4.19799837391576246663386741899, 5.23035060020067689268665763590, 5.74240001366221175143832145237, 6.22827256931015434938578234610, 7.53016255729802160081509384691, 7.77593179438390054962450948682