L(s) = 1 | + (2.27 − 1.68i)2-s + (2.33 − 7.65i)4-s + 4.43i·5-s + 7i·7-s + (−7.56 − 21.3i)8-s + (7.46 + 10.0i)10-s + 56.3·11-s + 57.7·13-s + (11.7 + 15.9i)14-s + (−53.0 − 35.7i)16-s − 109. i·17-s − 124. i·19-s + (33.9 + 10.3i)20-s + (128. − 94.8i)22-s − 156.·23-s + ⋯ |
L(s) = 1 | + (0.803 − 0.594i)2-s + (0.291 − 0.956i)4-s + 0.396i·5-s + 0.377i·7-s + (−0.334 − 0.942i)8-s + (0.235 + 0.318i)10-s + 1.54·11-s + 1.23·13-s + (0.224 + 0.303i)14-s + (−0.829 − 0.558i)16-s − 1.56i·17-s − 1.49i·19-s + (0.379 + 0.115i)20-s + (1.24 − 0.918i)22-s − 1.41·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.080509299\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.080509299\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.27 + 1.68i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
good | 5 | \( 1 - 4.43iT - 125T^{2} \) |
| 11 | \( 1 - 56.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 57.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 109. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 124. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 156.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 210. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 120. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 299.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 358. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 246. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 196.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 58.3iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 225.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 481.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 486. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 709.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 866.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 94.6iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 208.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.13e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.25e3T + 9.12e5T^{2} \) |
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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
−11.46405277454397005145686896667, −10.84702942980209616320625525863, −9.517760172385551007031630513575, −8.849922847043546144430507949802, −6.98144406116012012769450873138, −6.31917483654728485039538613002, −5.03713170202579518495109849214, −3.83912682228541329896255581816, −2.71257304980096145088170729943, −1.11391169127287914140971669826,
1.58529953252552486511890049084, 3.77026566572415111628057928181, 4.18625371595229882269218432182, 6.03570085548785738988011719763, 6.30698405251236628775962934270, 7.928449353493023052104059204865, 8.502737540518621557397505534609, 9.822842769821283032331025296793, 11.13767636357025318254150990109, 11.98662726304470068107578980889