L(s) = 1 | + 1.84·3-s − 3.37·5-s − 1.41·7-s + 0.414·9-s − 0.317·11-s + 1.84·13-s − 6.24·15-s + 2.82·17-s + 5.99·19-s − 2.61·21-s + 0.242·23-s + 6.41·25-s − 4.77·27-s − 2.93·29-s + 4·31-s − 0.585·33-s + 4.77·35-s − 1.84·37-s + 3.41·39-s − 8.24·41-s + 8.60·43-s − 1.39·45-s − 11.6·47-s − 5·49-s + 5.22·51-s − 8.15·53-s + 1.07·55-s + ⋯ |
L(s) = 1 | + 1.06·3-s − 1.51·5-s − 0.534·7-s + 0.138·9-s − 0.0955·11-s + 0.512·13-s − 1.61·15-s + 0.685·17-s + 1.37·19-s − 0.570·21-s + 0.0505·23-s + 1.28·25-s − 0.919·27-s − 0.544·29-s + 0.718·31-s − 0.101·33-s + 0.807·35-s − 0.303·37-s + 0.546·39-s − 1.28·41-s + 1.31·43-s − 0.208·45-s − 1.70·47-s − 0.714·49-s + 0.731·51-s − 1.12·53-s + 0.144·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 3 | \( 1 - 1.84T + 3T^{2} \) |
| 5 | \( 1 + 3.37T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 0.317T + 11T^{2} \) |
| 13 | \( 1 - 1.84T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 5.99T + 19T^{2} \) |
| 23 | \( 1 - 0.242T + 23T^{2} \) |
| 29 | \( 1 + 2.93T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 1.84T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 - 8.60T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 8.15T + 53T^{2} \) |
| 59 | \( 1 + 6.62T + 59T^{2} \) |
| 61 | \( 1 - 0.765T + 61T^{2} \) |
| 67 | \( 1 - 4.01T + 67T^{2} \) |
| 71 | \( 1 - 0.242T + 71T^{2} \) |
| 73 | \( 1 + 9.89T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 6.62T + 83T^{2} \) |
| 89 | \( 1 + 3.75T + 89T^{2} \) |
| 97 | \( 1 - 1.51T + 97T^{2} \) |
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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
−8.050482888927204104177090734767, −7.59322963499702983575274646762, −6.87046057610631399615488594753, −5.86286955903026277753706581164, −4.89980640084712586872231038132, −3.92985835947114477429553093006, −3.28803327478867586933481297161, −2.94277213532426302805159272212, −1.41548776858370716236933326977, 0,
1.41548776858370716236933326977, 2.94277213532426302805159272212, 3.28803327478867586933481297161, 3.92985835947114477429553093006, 4.89980640084712586872231038132, 5.86286955903026277753706581164, 6.87046057610631399615488594753, 7.59322963499702983575274646762, 8.050482888927204104177090734767