| L(s) = 1 | − 2.80·3-s − 3.33·5-s − 2.85·7-s + 4.87·9-s − 6.16·11-s − 3.18·13-s + 9.34·15-s − 3.44·17-s + 6.59·19-s + 8.01·21-s + 9.22·23-s + 6.09·25-s − 5.27·27-s + 1.40·29-s + 0.702·31-s + 17.2·33-s + 9.51·35-s − 9.54·37-s + 8.93·39-s − 2.82·41-s + 8.28·43-s − 16.2·45-s + 10.2·47-s + 1.16·49-s + 9.67·51-s − 5.03·53-s + 20.5·55-s + ⋯ |
| L(s) = 1 | − 1.62·3-s − 1.48·5-s − 1.07·7-s + 1.62·9-s − 1.85·11-s − 0.882·13-s + 2.41·15-s − 0.836·17-s + 1.51·19-s + 1.75·21-s + 1.92·23-s + 1.21·25-s − 1.01·27-s + 0.260·29-s + 0.126·31-s + 3.01·33-s + 1.60·35-s − 1.56·37-s + 1.43·39-s − 0.441·41-s + 1.26·43-s − 2.42·45-s + 1.49·47-s + 0.166·49-s + 1.35·51-s − 0.691·53-s + 2.76·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 + 3.33T + 5T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 11 | \( 1 + 6.16T + 11T^{2} \) |
| 13 | \( 1 + 3.18T + 13T^{2} \) |
| 17 | \( 1 + 3.44T + 17T^{2} \) |
| 19 | \( 1 - 6.59T + 19T^{2} \) |
| 23 | \( 1 - 9.22T + 23T^{2} \) |
| 29 | \( 1 - 1.40T + 29T^{2} \) |
| 31 | \( 1 - 0.702T + 31T^{2} \) |
| 37 | \( 1 + 9.54T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 - 8.28T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 5.03T + 53T^{2} \) |
| 59 | \( 1 + 3.57T + 59T^{2} \) |
| 61 | \( 1 - 2.46T + 61T^{2} \) |
| 67 | \( 1 - 0.271T + 67T^{2} \) |
| 71 | \( 1 + 6.18T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 7.67T + 79T^{2} \) |
| 83 | \( 1 - 1.85T + 83T^{2} \) |
| 89 | \( 1 + 1.46T + 89T^{2} \) |
| 97 | \( 1 + 0.763T + 97T^{2} \) |
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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
−7.71990123367816745430725870787, −7.22095531024185938872389316930, −6.79895615070678036543926888570, −5.68704802840728326145968197265, −5.06371066176856267589244837968, −4.60220662969913894105608256157, −3.41224446857942583122693095747, −2.71606524367283431925056571942, −0.71574843953701021542486200872, 0,
0.71574843953701021542486200872, 2.71606524367283431925056571942, 3.41224446857942583122693095747, 4.60220662969913894105608256157, 5.06371066176856267589244837968, 5.68704802840728326145968197265, 6.79895615070678036543926888570, 7.22095531024185938872389316930, 7.71990123367816745430725870787
