L(s) = 1 | + 2.37·3-s − 0.377·7-s + 2.65·9-s + 1.37·11-s − 2.82·13-s − 6.37·17-s − 19-s − 0.896·21-s + 6.19·23-s − 0.829·27-s − 3.37·29-s − 2.48·31-s + 3.27·33-s − 5.58·37-s − 6.70·39-s + 8.50·41-s − 12.1·43-s − 6.87·47-s − 6.85·49-s − 15.1·51-s − 11.5·53-s − 2.37·57-s − 6.05·59-s + 5.02·61-s − 63-s + 3.22·67-s + 14.7·69-s + ⋯ |
L(s) = 1 | + 1.37·3-s − 0.142·7-s + 0.883·9-s + 0.415·11-s − 0.782·13-s − 1.54·17-s − 0.229·19-s − 0.195·21-s + 1.29·23-s − 0.159·27-s − 0.627·29-s − 0.445·31-s + 0.569·33-s − 0.917·37-s − 1.07·39-s + 1.32·41-s − 1.85·43-s − 1.00·47-s − 0.979·49-s − 2.12·51-s − 1.58·53-s − 0.314·57-s − 0.788·59-s + 0.643·61-s − 0.125·63-s + 0.394·67-s + 1.77·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 2.37T + 3T^{2} \) |
| 7 | \( 1 + 0.377T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 6.37T + 17T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 + 3.37T + 29T^{2} \) |
| 31 | \( 1 + 2.48T + 31T^{2} \) |
| 37 | \( 1 + 5.58T + 37T^{2} \) |
| 41 | \( 1 - 8.50T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 + 6.87T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 6.05T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 - 3.22T + 67T^{2} \) |
| 71 | \( 1 - 2.30T + 71T^{2} \) |
| 73 | \( 1 - 3.19T + 73T^{2} \) |
| 79 | \( 1 - 6.71T + 79T^{2} \) |
| 83 | \( 1 - 18.2T + 83T^{2} \) |
| 89 | \( 1 - 1.50T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.70543837314369368154596317897, −6.81494020797618312300308720666, −6.50499032980325763257116174955, −5.21096926983598594365003304519, −4.62572684522604573198777321625, −3.71998442205173957227443118014, −3.11632180715378391869153508418, −2.30441746569013997102147899244, −1.63657300475069191561447173639, 0,
1.63657300475069191561447173639, 2.30441746569013997102147899244, 3.11632180715378391869153508418, 3.71998442205173957227443118014, 4.62572684522604573198777321625, 5.21096926983598594365003304519, 6.50499032980325763257116174955, 6.81494020797618312300308720666, 7.70543837314369368154596317897