L(s) = 1 | − 3-s + 5-s + 9-s + 6·11-s + 4·13-s − 15-s − 6·17-s − 2·19-s + 25-s − 27-s + 6·29-s + 10·31-s − 6·33-s + 2·37-s − 4·39-s + 6·41-s − 4·43-s + 45-s + 6·51-s − 12·53-s + 6·55-s + 2·57-s − 14·61-s + 4·65-s − 4·67-s + 6·71-s + 4·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.80·11-s + 1.10·13-s − 0.258·15-s − 1.45·17-s − 0.458·19-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.79·31-s − 1.04·33-s + 0.328·37-s − 0.640·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s + 0.840·51-s − 1.64·53-s + 0.809·55-s + 0.264·57-s − 1.79·61-s + 0.496·65-s − 0.488·67-s + 0.712·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.934750632\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.934750632\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.853676270417667680852168008738, −8.150601586294785775851370103484, −6.89699202824220487199225928595, −6.26979462967166440860357968043, −6.14184640813796171219824287672, −4.63672703414062759217241031160, −4.29417620212795259697627947254, −3.14485807783240587800268884275, −1.85300873598198939037163807777, −0.939689875495428387669788785603,
0.939689875495428387669788785603, 1.85300873598198939037163807777, 3.14485807783240587800268884275, 4.29417620212795259697627947254, 4.63672703414062759217241031160, 6.14184640813796171219824287672, 6.26979462967166440860357968043, 6.89699202824220487199225928595, 8.150601586294785775851370103484, 8.853676270417667680852168008738