| L(s) = 1 | − 3-s − 5-s + 9-s + 2·13-s + 15-s − 6·17-s + 4·19-s + 25-s − 27-s + 6·29-s + 8·31-s − 2·37-s − 2·39-s + 6·41-s − 4·43-s − 45-s + 6·51-s + 6·53-s − 4·57-s − 10·61-s − 2·65-s − 4·67-s − 2·73-s − 75-s − 8·79-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.554·13-s + 0.258·15-s − 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 0.840·51-s + 0.824·53-s − 0.529·57-s − 1.28·61-s − 0.248·65-s − 0.488·67-s − 0.234·73-s − 0.115·75-s − 0.900·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.02615457909162, −14.20157920362234, −13.79711883406241, −13.32000708122529, −12.76949093927536, −12.18261280292159, −11.61412064092703, −11.43320008357128, −10.69505272492021, −10.31800612454936, −9.710420420523697, −8.991024351177508, −8.589057095722742, −7.979657536255656, −7.372739015934719, −6.716199403046334, −6.401991248595934, −5.687846410876592, −5.073812769059882, −4.338925100913110, −4.146168499095642, −3.083805564556142, −2.648181815811365, −1.585843199716292, −0.9102178734169543, 0,
0.9102178734169543, 1.585843199716292, 2.648181815811365, 3.083805564556142, 4.146168499095642, 4.338925100913110, 5.073812769059882, 5.687846410876592, 6.401991248595934, 6.716199403046334, 7.372739015934719, 7.979657536255656, 8.589057095722742, 8.991024351177508, 9.710420420523697, 10.31800612454936, 10.69505272492021, 11.43320008357128, 11.61412064092703, 12.18261280292159, 12.76949093927536, 13.32000708122529, 13.79711883406241, 14.20157920362234, 15.02615457909162
