L(s) = 1 | + 2-s + 3-s + 4-s − 2.78·5-s + 6-s + 0.966·7-s + 8-s + 9-s − 2.78·10-s − 11-s + 12-s − 0.856·13-s + 0.966·14-s − 2.78·15-s + 16-s − 2.44·17-s + 18-s − 6.19·19-s − 2.78·20-s + 0.966·21-s − 22-s + 3.52·23-s + 24-s + 2.75·25-s − 0.856·26-s + 27-s + 0.966·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.24·5-s + 0.408·6-s + 0.365·7-s + 0.353·8-s + 0.333·9-s − 0.880·10-s − 0.301·11-s + 0.288·12-s − 0.237·13-s + 0.258·14-s − 0.718·15-s + 0.250·16-s − 0.593·17-s + 0.235·18-s − 1.42·19-s − 0.622·20-s + 0.210·21-s − 0.213·22-s + 0.735·23-s + 0.204·24-s + 0.550·25-s − 0.168·26-s + 0.192·27-s + 0.182·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 5 | \( 1 + 2.78T + 5T^{2} \) |
| 7 | \( 1 - 0.966T + 7T^{2} \) |
| 13 | \( 1 + 0.856T + 13T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 + 6.19T + 19T^{2} \) |
| 23 | \( 1 - 3.52T + 23T^{2} \) |
| 29 | \( 1 - 0.0503T + 29T^{2} \) |
| 31 | \( 1 + 8.57T + 31T^{2} \) |
| 37 | \( 1 + 1.77T + 37T^{2} \) |
| 41 | \( 1 + 0.392T + 41T^{2} \) |
| 43 | \( 1 - 4.20T + 43T^{2} \) |
| 47 | \( 1 + 5.92T + 47T^{2} \) |
| 53 | \( 1 + 4.08T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 67 | \( 1 - 4.74T + 67T^{2} \) |
| 71 | \( 1 + 9.26T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 5.02T + 83T^{2} \) |
| 89 | \( 1 - 7.24T + 89T^{2} \) |
| 97 | \( 1 + 7.72T + 97T^{2} \) |
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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
−8.102261951672565586849364353537, −7.26408397998693664476723420155, −6.83933440468209262845648281531, −5.74513808648770347739295810977, −4.78445908752774958921417645365, −4.23780504790073482663480606576, −3.54952864694099596095398532760, −2.66971518362161352447103546641, −1.71436565782694536087908154590, 0,
1.71436565782694536087908154590, 2.66971518362161352447103546641, 3.54952864694099596095398532760, 4.23780504790073482663480606576, 4.78445908752774958921417645365, 5.74513808648770347739295810977, 6.83933440468209262845648281531, 7.26408397998693664476723420155, 8.102261951672565586849364353537