L(s) = 1 | − 2.30·3-s + 2.18·5-s − 0.264·7-s + 2.33·9-s − 0.777·11-s + 3.18·13-s − 5.03·15-s + 17-s − 2.74·19-s + 0.610·21-s − 2.84·23-s − 0.237·25-s + 1.54·27-s − 2.79·29-s − 7.75·31-s + 1.79·33-s − 0.576·35-s + 1.58·37-s − 7.35·39-s + 8.80·41-s − 6.38·43-s + 5.09·45-s + 9.16·47-s − 6.93·49-s − 2.30·51-s − 11.2·53-s − 1.69·55-s + ⋯ |
L(s) = 1 | − 1.33·3-s + 0.976·5-s − 0.0998·7-s + 0.777·9-s − 0.234·11-s + 0.883·13-s − 1.30·15-s + 0.242·17-s − 0.629·19-s + 0.133·21-s − 0.593·23-s − 0.0474·25-s + 0.296·27-s − 0.518·29-s − 1.39·31-s + 0.312·33-s − 0.0974·35-s + 0.260·37-s − 1.17·39-s + 1.37·41-s − 0.973·43-s + 0.759·45-s + 1.33·47-s − 0.990·49-s − 0.323·51-s − 1.54·53-s − 0.228·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 - 2.18T + 5T^{2} \) |
| 7 | \( 1 + 0.264T + 7T^{2} \) |
| 11 | \( 1 + 0.777T + 11T^{2} \) |
| 13 | \( 1 - 3.18T + 13T^{2} \) |
| 19 | \( 1 + 2.74T + 19T^{2} \) |
| 23 | \( 1 + 2.84T + 23T^{2} \) |
| 29 | \( 1 + 2.79T + 29T^{2} \) |
| 31 | \( 1 + 7.75T + 31T^{2} \) |
| 37 | \( 1 - 1.58T + 37T^{2} \) |
| 41 | \( 1 - 8.80T + 41T^{2} \) |
| 43 | \( 1 + 6.38T + 43T^{2} \) |
| 47 | \( 1 - 9.16T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 61 | \( 1 - 5.95T + 61T^{2} \) |
| 67 | \( 1 - 0.642T + 67T^{2} \) |
| 71 | \( 1 + 5.12T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 + 6.54T + 79T^{2} \) |
| 83 | \( 1 + 4.95T + 83T^{2} \) |
| 89 | \( 1 + 1.86T + 89T^{2} \) |
| 97 | \( 1 + 3.35T + 97T^{2} \) |
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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
−8.039493275949453156736035134424, −7.14423999998533712923756525036, −6.25383821573820289627112256000, −5.92731448686403489941511397033, −5.36269442745025930549625019259, −4.46624234138144711072289324044, −3.51565309864571985396460625955, −2.24864574689410244526664039045, −1.32199445430100795155136140779, 0,
1.32199445430100795155136140779, 2.24864574689410244526664039045, 3.51565309864571985396460625955, 4.46624234138144711072289324044, 5.36269442745025930549625019259, 5.92731448686403489941511397033, 6.25383821573820289627112256000, 7.14423999998533712923756525036, 8.039493275949453156736035134424