| L(s) = 1 | + 3-s − 2·7-s + 9-s + 2·13-s + 2·19-s − 2·21-s + 27-s − 8·31-s − 2·37-s + 2·39-s − 2·43-s − 3·49-s − 6·53-s + 2·57-s + 12·59-s − 2·61-s − 2·63-s − 4·67-s + 2·73-s − 10·79-s + 81-s + 12·83-s − 6·89-s − 4·91-s − 8·93-s − 14·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 0.458·19-s − 0.436·21-s + 0.192·27-s − 1.43·31-s − 0.328·37-s + 0.320·39-s − 0.304·43-s − 3/7·49-s − 0.824·53-s + 0.264·57-s + 1.56·59-s − 0.256·61-s − 0.251·63-s − 0.488·67-s + 0.234·73-s − 1.12·79-s + 1/9·81-s + 1.31·83-s − 0.635·89-s − 0.419·91-s − 0.829·93-s − 1.42·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.072847235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.072847235\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.33964026440741, −12.83892590364736, −12.70633154445563, −11.92403479029060, −11.50917999171754, −10.91368221913453, −10.50061469910825, −9.802954476220110, −9.635115746507892, −8.896209308003431, −8.726739786363496, −7.958087580215987, −7.598619635397773, −6.921398416517917, −6.618246190026019, −5.937978446290984, −5.446163199138960, −4.872041839388889, −4.111372507773400, −3.648838121443759, −3.197744844708633, −2.640358133560670, −1.875414729045129, −1.327265254477163, −0.4083881999744354,
0.4083881999744354, 1.327265254477163, 1.875414729045129, 2.640358133560670, 3.197744844708633, 3.648838121443759, 4.111372507773400, 4.872041839388889, 5.446163199138960, 5.937978446290984, 6.618246190026019, 6.921398416517917, 7.598619635397773, 7.958087580215987, 8.726739786363496, 8.896209308003431, 9.635115746507892, 9.802954476220110, 10.50061469910825, 10.91368221913453, 11.50917999171754, 11.92403479029060, 12.70633154445563, 12.83892590364736, 13.33964026440741
