L(s) = 1 | + i·3-s + 2.47i·5-s − 2.55·7-s − 9-s + 0.669i·11-s − 4.08i·13-s − 2.47·15-s + 6.44·17-s − 6.44i·19-s − 2.55i·21-s + 2.82·23-s − 1.11·25-s − i·27-s − 4.35i·29-s − 6.55·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.10i·5-s − 0.966·7-s − 0.333·9-s + 0.201i·11-s − 1.13i·13-s − 0.638·15-s + 1.56·17-s − 1.47i·19-s − 0.558i·21-s + 0.589·23-s − 0.223·25-s − 0.192i·27-s − 0.808i·29-s − 1.17·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.427841727\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.427841727\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
good | 5 | \( 1 - 2.47iT - 5T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 - 0.669iT - 11T^{2} \) |
| 13 | \( 1 + 4.08iT - 13T^{2} \) |
| 17 | \( 1 - 6.44T + 17T^{2} \) |
| 19 | \( 1 + 6.44iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 4.35iT - 29T^{2} \) |
| 31 | \( 1 + 6.55T + 31T^{2} \) |
| 37 | \( 1 + 3.85iT - 37T^{2} \) |
| 41 | \( 1 + 0.788T + 41T^{2} \) |
| 43 | \( 1 + 0.550iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 3.64iT - 53T^{2} \) |
| 59 | \( 1 + 5.65iT - 59T^{2} \) |
| 61 | \( 1 + 6.20iT - 61T^{2} \) |
| 67 | \( 1 - 2.99iT - 67T^{2} \) |
| 71 | \( 1 - 5.11T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 6.31T + 79T^{2} \) |
| 83 | \( 1 - 0.907iT - 83T^{2} \) |
| 89 | \( 1 - 6.31T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−8.864044449426771805751739969976, −7.80143472639530778533690494164, −7.21104565495802094867417739106, −6.45344605186764234458029706025, −5.66690120220221941689534838957, −4.94839457235190040967003709317, −3.64390511414266871370337933563, −3.19439322986092907175846329698, −2.46416384824110385239545507976, −0.55100458888902263295292373052,
0.976980473448459421816177034803, 1.79164628328548991378238026534, 3.19171548592230284062624845366, 3.85470015373742224190889269354, 5.01300175062752681504219918948, 5.67879887776666989884640320044, 6.43112059276035337046748844864, 7.22731956880204589860313973094, 7.980493157110369153245819827599, 8.744884202914717085420417432723