| L(s) = 1 | − 2.44·3-s + 3.44·5-s + 7-s + 2.99·9-s − 6.44·11-s − 8.44·15-s + 2.44·17-s − 0.550·19-s − 2.44·21-s + 3.89·23-s + 6.89·25-s − 5.89·29-s − 3.44·31-s + 15.7·33-s + 3.44·35-s + 8.44·37-s − 10.8·41-s + 43-s + 10.3·45-s − 3.44·47-s + 49-s − 5.99·51-s + 1.89·53-s − 22.2·55-s + 1.34·57-s − 14·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 1.41·3-s + 1.54·5-s + 0.377·7-s + 0.999·9-s − 1.94·11-s − 2.18·15-s + 0.594·17-s − 0.126·19-s − 0.534·21-s + 0.812·23-s + 1.37·25-s − 1.09·29-s − 0.619·31-s + 2.75·33-s + 0.583·35-s + 1.38·37-s − 1.70·41-s + 0.152·43-s + 1.54·45-s − 0.503·47-s + 0.142·49-s − 0.840·51-s + 0.260·53-s − 2.99·55-s + 0.178·57-s − 1.82·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4732 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4732 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 - 3.44T + 5T^{2} \) |
| 11 | \( 1 + 6.44T + 11T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 + 0.550T + 19T^{2} \) |
| 23 | \( 1 - 3.89T + 23T^{2} \) |
| 29 | \( 1 + 5.89T + 29T^{2} \) |
| 31 | \( 1 + 3.44T + 31T^{2} \) |
| 37 | \( 1 - 8.44T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + 3.44T + 47T^{2} \) |
| 53 | \( 1 - 1.89T + 53T^{2} \) |
| 59 | \( 1 + 14T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 6.89T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 2.10T + 79T^{2} \) |
| 83 | \( 1 - 4.34T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−7.75395021130058667376319997148, −7.12061691987041034387896991583, −6.09739354044974815026020604864, −5.77496711167120305668973535158, −5.11058702668511010964885646170, −4.75776695921965735760172273127, −3.15902455871694063621466031184, −2.24642756074403717755709477469, −1.32080374481205256579412918217, 0,
1.32080374481205256579412918217, 2.24642756074403717755709477469, 3.15902455871694063621466031184, 4.75776695921965735760172273127, 5.11058702668511010964885646170, 5.77496711167120305668973535158, 6.09739354044974815026020604864, 7.12061691987041034387896991583, 7.75395021130058667376319997148
