| L(s) = 1 | + 6·11-s − 13-s − 2·17-s + 4·23-s − 5·25-s + 6·29-s + 4·31-s − 2·37-s − 4·43-s + 10·47-s − 7·49-s + 10·53-s − 6·59-s − 6·61-s + 12·67-s + 2·71-s + 6·73-s + 16·79-s + 6·83-s − 4·89-s + 14·97-s + 6·101-s + 8·103-s − 8·107-s − 6·109-s + 10·113-s + ⋯ |
| L(s) = 1 | + 1.80·11-s − 0.277·13-s − 0.485·17-s + 0.834·23-s − 25-s + 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.609·43-s + 1.45·47-s − 49-s + 1.37·53-s − 0.781·59-s − 0.768·61-s + 1.46·67-s + 0.237·71-s + 0.702·73-s + 1.80·79-s + 0.658·83-s − 0.423·89-s + 1.42·97-s + 0.597·101-s + 0.788·103-s − 0.773·107-s − 0.574·109-s + 0.940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.917489087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.917489087\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.184294838662426330599174276496, −8.603027162415544040212315107506, −7.63584831683610361882108019767, −6.69087411967754787433668721392, −6.28943189296774299916363973955, −5.09555003610369654280744252248, −4.24486360092282550965285188483, −3.43299310554595416240944294936, −2.19080394870072878544535794019, −0.986325425668759395181242182793,
0.986325425668759395181242182793, 2.19080394870072878544535794019, 3.43299310554595416240944294936, 4.24486360092282550965285188483, 5.09555003610369654280744252248, 6.28943189296774299916363973955, 6.69087411967754787433668721392, 7.63584831683610361882108019767, 8.603027162415544040212315107506, 9.184294838662426330599174276496
