| L(s) = 1 | + 0.687·3-s − 1.01·7-s − 2.52·9-s − 5.12·11-s + 6.08·13-s + 3.19·17-s − 3.42·19-s − 0.695·21-s − 2.91·23-s − 3.79·27-s − 1.55·29-s + 7.99·31-s − 3.52·33-s + 8.40·37-s + 4.18·39-s − 1.86·41-s + 5.22·43-s + 4.80·47-s − 5.97·49-s + 2.19·51-s + 10.0·53-s − 2.35·57-s − 2.89·59-s − 2.30·61-s + 2.55·63-s − 4.64·67-s − 2.00·69-s + ⋯ |
| L(s) = 1 | + 0.396·3-s − 0.382·7-s − 0.842·9-s − 1.54·11-s + 1.68·13-s + 0.774·17-s − 0.786·19-s − 0.151·21-s − 0.608·23-s − 0.731·27-s − 0.288·29-s + 1.43·31-s − 0.612·33-s + 1.38·37-s + 0.669·39-s − 0.291·41-s + 0.796·43-s + 0.700·47-s − 0.853·49-s + 0.307·51-s + 1.38·53-s − 0.312·57-s − 0.376·59-s − 0.295·61-s + 0.322·63-s − 0.567·67-s − 0.241·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.687T + 3T^{2} \) |
| 7 | \( 1 + 1.01T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 - 6.08T + 13T^{2} \) |
| 17 | \( 1 - 3.19T + 17T^{2} \) |
| 19 | \( 1 + 3.42T + 19T^{2} \) |
| 23 | \( 1 + 2.91T + 23T^{2} \) |
| 29 | \( 1 + 1.55T + 29T^{2} \) |
| 31 | \( 1 - 7.99T + 31T^{2} \) |
| 37 | \( 1 - 8.40T + 37T^{2} \) |
| 41 | \( 1 + 1.86T + 41T^{2} \) |
| 43 | \( 1 - 5.22T + 43T^{2} \) |
| 47 | \( 1 - 4.80T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 2.89T + 59T^{2} \) |
| 61 | \( 1 + 2.30T + 61T^{2} \) |
| 67 | \( 1 + 4.64T + 67T^{2} \) |
| 71 | \( 1 - 7.73T + 71T^{2} \) |
| 73 | \( 1 + 0.595T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 6.39T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−7.52096421355458885993366843022, −6.49388306586255228482897890258, −5.87091022129404020410633835317, −5.52368610229337966213529721496, −4.41945311160475458020152011989, −3.72390998468412674001715929306, −2.88020911749532139646458369975, −2.45405030149670970491285088307, −1.18620121442650021514950889807, 0,
1.18620121442650021514950889807, 2.45405030149670970491285088307, 2.88020911749532139646458369975, 3.72390998468412674001715929306, 4.41945311160475458020152011989, 5.52368610229337966213529721496, 5.87091022129404020410633835317, 6.49388306586255228482897890258, 7.52096421355458885993366843022
