| L(s) = 1 | − 2-s + 0.516·3-s + 4-s − 1.53·5-s − 0.516·6-s + 3.77·7-s − 8-s − 2.73·9-s + 1.53·10-s + 1.38·11-s + 0.516·12-s − 3.77·14-s − 0.791·15-s + 16-s + 6.33·17-s + 2.73·18-s + 19-s − 1.53·20-s + 1.94·21-s − 1.38·22-s − 1.87·23-s − 0.516·24-s − 2.65·25-s − 2.96·27-s + 3.77·28-s − 1.81·29-s + 0.791·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.298·3-s + 0.5·4-s − 0.685·5-s − 0.210·6-s + 1.42·7-s − 0.353·8-s − 0.911·9-s + 0.484·10-s + 0.416·11-s + 0.149·12-s − 1.00·14-s − 0.204·15-s + 0.250·16-s + 1.53·17-s + 0.644·18-s + 0.229·19-s − 0.342·20-s + 0.425·21-s − 0.294·22-s − 0.391·23-s − 0.105·24-s − 0.530·25-s − 0.569·27-s + 0.712·28-s − 0.336·29-s + 0.144·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.516T + 3T^{2} \) |
| 5 | \( 1 + 1.53T + 5T^{2} \) |
| 7 | \( 1 - 3.77T + 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 17 | \( 1 - 6.33T + 17T^{2} \) |
| 23 | \( 1 + 1.87T + 23T^{2} \) |
| 29 | \( 1 + 1.81T + 29T^{2} \) |
| 31 | \( 1 + 7.35T + 31T^{2} \) |
| 37 | \( 1 + 3.38T + 37T^{2} \) |
| 41 | \( 1 + 7.60T + 41T^{2} \) |
| 43 | \( 1 + 7.68T + 43T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 2.40T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 2.95T + 67T^{2} \) |
| 71 | \( 1 - 8.47T + 71T^{2} \) |
| 73 | \( 1 - 1.17T + 73T^{2} \) |
| 79 | \( 1 - 1.27T + 79T^{2} \) |
| 83 | \( 1 - 8.99T + 83T^{2} \) |
| 89 | \( 1 - 1.41T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−7.85791328060043014816970824567, −7.40331154811641990309052819065, −6.32724568537224589058490103344, −5.48984222536698717206487894095, −4.93181957244916318833138847093, −3.74932325334194751822793537332, −3.27585520582534809588333660917, −2.04233893832482255958666034402, −1.35445357624630814529047050194, 0,
1.35445357624630814529047050194, 2.04233893832482255958666034402, 3.27585520582534809588333660917, 3.74932325334194751822793537332, 4.93181957244916318833138847093, 5.48984222536698717206487894095, 6.32724568537224589058490103344, 7.40331154811641990309052819065, 7.85791328060043014816970824567
