L(s) = 1 | − 2i·3-s − i·7-s − 9-s − 5·11-s + i·13-s + 4i·17-s − 7·19-s − 2·21-s − i·23-s − 4i·27-s − 5·29-s + 2·31-s + 10i·33-s + 2i·37-s + 2·39-s + ⋯ |
L(s) = 1 | − 1.15i·3-s − 0.377i·7-s − 0.333·9-s − 1.50·11-s + 0.277i·13-s + 0.970i·17-s − 1.60·19-s − 0.436·21-s − 0.208i·23-s − 0.769i·27-s − 0.928·29-s + 0.359·31-s + 1.74i·33-s + 0.328i·37-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9853730257\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9853730257\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 11T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 14T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 + 7T + 79T^{2} \) |
| 83 | \( 1 + 15iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 4iT - 97T^{2} \) |
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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
−8.264269131570749903758974397380, −7.55609953443471149342391748458, −7.09066014002348235488003929050, −6.20050033256577282849210668107, −5.73041631065807757287145543299, −4.57437683728827654822942559735, −3.92718986990685228776643370656, −2.58279445913772675595450311801, −2.07690155642178728129418253807, −0.927615109624252587194619249141,
0.31162210229232758839415089723, 2.16377438635815451101202007456, 2.83057015664671695292213197785, 3.86423869656204061429269847821, 4.51207667870896595434632516649, 5.34843722596875682120949105119, 5.70166997028613125475032034557, 6.89796241843628698252813327936, 7.59778837321851120194824909342, 8.424787667637716630155467492977