L(s) = 1 | + 5-s − 13-s − 2·17-s − 4·23-s + 25-s − 6·29-s + 8·31-s + 10·37-s + 6·41-s + 4·43-s + 8·47-s − 7·49-s − 2·53-s + 6·61-s − 65-s + 12·67-s − 8·71-s + 6·73-s + 8·79-s + 12·83-s − 2·85-s + 14·89-s − 10·97-s − 6·101-s + 4·103-s − 4·107-s − 6·109-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.277·13-s − 0.485·17-s − 0.834·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 1.64·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s − 49-s − 0.274·53-s + 0.768·61-s − 0.124·65-s + 1.46·67-s − 0.949·71-s + 0.702·73-s + 0.900·79-s + 1.31·83-s − 0.216·85-s + 1.48·89-s − 1.01·97-s − 0.597·101-s + 0.394·103-s − 0.386·107-s − 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.979095899\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.979095899\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.146531237861408885814809523576, −7.72966608688262107933963345171, −6.74408814663702631340045436167, −6.14905162881414262045771458102, −5.44481973279538650948175619708, −4.54663181458360238880338418119, −3.87032996231249797962596787055, −2.71252329069690914005496453249, −2.05358769457321749870469485946, −0.77562474269981314412438783142,
0.77562474269981314412438783142, 2.05358769457321749870469485946, 2.71252329069690914005496453249, 3.87032996231249797962596787055, 4.54663181458360238880338418119, 5.44481973279538650948175619708, 6.14905162881414262045771458102, 6.74408814663702631340045436167, 7.72966608688262107933963345171, 8.146531237861408885814809523576