| L(s) = 1 | + 3-s − 5-s + 9-s − 4·11-s + 13-s − 15-s + 6·17-s − 4·19-s + 25-s + 27-s + 2·29-s − 8·31-s − 4·33-s + 6·37-s + 39-s − 10·41-s + 4·43-s − 45-s − 8·47-s − 7·49-s + 6·51-s + 6·53-s + 4·55-s − 4·57-s + 12·59-s − 10·61-s − 65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.258·15-s + 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.696·33-s + 0.986·37-s + 0.160·39-s − 1.56·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s − 49-s + 0.840·51-s + 0.824·53-s + 0.539·55-s − 0.529·57-s + 1.56·59-s − 1.28·61-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.77570422040689307913155013041, −7.22407121065001314998792863678, −6.30669946708399073289919407171, −5.47284982755510309041547227069, −4.81511853136599012806047109161, −3.87028623304558581738204600257, −3.22807811916170197570085757388, −2.43846841180612958383104846332, −1.38864283232714263424235202368, 0,
1.38864283232714263424235202368, 2.43846841180612958383104846332, 3.22807811916170197570085757388, 3.87028623304558581738204600257, 4.81511853136599012806047109161, 5.47284982755510309041547227069, 6.30669946708399073289919407171, 7.22407121065001314998792863678, 7.77570422040689307913155013041
