| L(s) = 1 | − 2.41·3-s + 5-s + 2.82·9-s + 1.82·11-s − 6.41·13-s − 2.41·15-s − 3.58·17-s + 7.65·19-s + 3.41·23-s + 25-s + 0.414·27-s − 4.65·29-s − 7.41·31-s − 4.41·33-s − 0.585·37-s + 15.4·39-s − 3.41·41-s + 0.343·43-s + 2.82·45-s − 10.8·47-s + 8.65·51-s − 12.2·53-s + 1.82·55-s − 18.4·57-s − 0.585·59-s − 10.8·61-s − 6.41·65-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 0.447·5-s + 0.942·9-s + 0.551·11-s − 1.77·13-s − 0.623·15-s − 0.869·17-s + 1.75·19-s + 0.711·23-s + 0.200·25-s + 0.0797·27-s − 0.864·29-s − 1.33·31-s − 0.768·33-s − 0.0963·37-s + 2.47·39-s − 0.533·41-s + 0.0523·43-s + 0.421·45-s − 1.58·47-s + 1.21·51-s − 1.68·53-s + 0.246·55-s − 2.44·57-s − 0.0762·59-s − 1.38·61-s − 0.795·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 11 | \( 1 - 1.82T + 11T^{2} \) |
| 13 | \( 1 + 6.41T + 13T^{2} \) |
| 17 | \( 1 + 3.58T + 17T^{2} \) |
| 19 | \( 1 - 7.65T + 19T^{2} \) |
| 23 | \( 1 - 3.41T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 + 7.41T + 31T^{2} \) |
| 37 | \( 1 + 0.585T + 37T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 43 | \( 1 - 0.343T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 0.585T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 3.07T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 9.72T + 97T^{2} \) |
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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
−9.631238679195046345978330328254, −9.102691490295267057013913864365, −7.54095536878199713415402242985, −6.97627950513349276641135508582, −6.05815683316642060527335324338, −5.18558606489697503301758832989, −4.69839196630239099295080704262, −3.12718643944308874880015815755, −1.63176368754881663258788704565, 0,
1.63176368754881663258788704565, 3.12718643944308874880015815755, 4.69839196630239099295080704262, 5.18558606489697503301758832989, 6.05815683316642060527335324338, 6.97627950513349276641135508582, 7.54095536878199713415402242985, 9.102691490295267057013913864365, 9.631238679195046345978330328254
