L(s) = 1 | + 3·3-s + 6·9-s − 11-s + 2·13-s + 3·17-s − 5·19-s + 3·23-s + 9·27-s − 6·29-s + 31-s − 3·33-s + 5·37-s + 6·39-s + 10·41-s + 4·43-s + 47-s + 9·51-s + 9·53-s − 15·57-s − 3·59-s − 3·61-s − 11·67-s + 9·69-s + 16·71-s + 7·73-s − 11·79-s + 9·81-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 2·9-s − 0.301·11-s + 0.554·13-s + 0.727·17-s − 1.14·19-s + 0.625·23-s + 1.73·27-s − 1.11·29-s + 0.179·31-s − 0.522·33-s + 0.821·37-s + 0.960·39-s + 1.56·41-s + 0.609·43-s + 0.145·47-s + 1.26·51-s + 1.23·53-s − 1.98·57-s − 0.390·59-s − 0.384·61-s − 1.34·67-s + 1.08·69-s + 1.89·71-s + 0.819·73-s − 1.23·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.506150523\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.506150523\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−7.59246023884489711974266453849, −7.44540340002248345637259370055, −6.37996337941265309064798494154, −5.70123273648231462999902278951, −4.63867966620676012369169787758, −3.98484270876123289992019479950, −3.37762912695509789200650090415, −2.59538379559023374004578878668, −2.00062890680846465121119341997, −0.950579576970159971938950046175,
0.950579576970159971938950046175, 2.00062890680846465121119341997, 2.59538379559023374004578878668, 3.37762912695509789200650090415, 3.98484270876123289992019479950, 4.63867966620676012369169787758, 5.70123273648231462999902278951, 6.37996337941265309064798494154, 7.44540340002248345637259370055, 7.59246023884489711974266453849