L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 2·13-s − 14-s + 16-s + 2·19-s − 20-s + 25-s − 2·26-s + 28-s + 6·29-s + 8·31-s − 32-s − 35-s − 4·37-s − 2·38-s + 40-s + 6·41-s + 2·43-s − 6·47-s + 49-s − 50-s + 2·52-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 + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.458·19-s − 0.223·20-s + 1/5·25-s − 0.392·26-s + 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.169·35-s − 0.657·37-s − 0.324·38-s + 0.158·40-s + 0.937·41-s + 0.304·43-s − 0.875·47-s + 1/7·49-s − 0.141·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.063854504\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.063854504\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.51368786694604441485994743120, −9.778776947950864046321352849397, −8.665263737968927740465692008259, −8.166330233724879768310807906987, −7.19144165603998610355864145787, −6.30379462467604097810240064468, −5.11297880694047186420183754307, −3.91020711853304460186990731425, −2.62844127442401831877194237389, −1.04459283013202278195173029970,
1.04459283013202278195173029970, 2.62844127442401831877194237389, 3.91020711853304460186990731425, 5.11297880694047186420183754307, 6.30379462467604097810240064468, 7.19144165603998610355864145787, 8.166330233724879768310807906987, 8.665263737968927740465692008259, 9.778776947950864046321352849397, 10.51368786694604441485994743120