L(s) = 1 | + 3-s − 0.618·5-s + 7-s + 9-s − 0.236·11-s + 0.381·13-s − 0.618·15-s − 2.23·17-s − 3.47·19-s + 21-s − 23-s − 4.61·25-s + 27-s + 5.47·29-s + 1.76·31-s − 0.236·33-s − 0.618·35-s − 5.47·37-s + 0.381·39-s + 7.47·41-s + 12.5·43-s − 0.618·45-s + 0.763·47-s + 49-s − 2.23·51-s − 7.09·53-s + 0.145·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.276·5-s + 0.377·7-s + 0.333·9-s − 0.0711·11-s + 0.105·13-s − 0.159·15-s − 0.542·17-s − 0.796·19-s + 0.218·21-s − 0.208·23-s − 0.923·25-s + 0.192·27-s + 1.01·29-s + 0.316·31-s − 0.0410·33-s − 0.104·35-s − 0.899·37-s + 0.0611·39-s + 1.16·41-s + 1.91·43-s − 0.0921·45-s + 0.111·47-s + 0.142·49-s − 0.313·51-s − 0.973·53-s + 0.0196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.355372576\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.355372576\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 0.618T + 5T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 - 0.381T + 13T^{2} \) |
| 17 | \( 1 + 2.23T + 17T^{2} \) |
| 19 | \( 1 + 3.47T + 19T^{2} \) |
| 29 | \( 1 - 5.47T + 29T^{2} \) |
| 31 | \( 1 - 1.76T + 31T^{2} \) |
| 37 | \( 1 + 5.47T + 37T^{2} \) |
| 41 | \( 1 - 7.47T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 0.763T + 47T^{2} \) |
| 53 | \( 1 + 7.09T + 53T^{2} \) |
| 59 | \( 1 - 3.38T + 59T^{2} \) |
| 61 | \( 1 - 5.32T + 61T^{2} \) |
| 67 | \( 1 - 3.38T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 1.94T + 73T^{2} \) |
| 79 | \( 1 - 8.23T + 79T^{2} \) |
| 83 | \( 1 - 7.94T + 83T^{2} \) |
| 89 | \( 1 + 2.85T + 89T^{2} \) |
| 97 | \( 1 + 0.236T + 97T^{2} \) |
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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
−7.994920884572981285922036248461, −7.28517638205992651476170625719, −6.52995576064869421005831147608, −5.85588904112827243672923891660, −4.88562457715919204178376484130, −4.23458362790248218519386092577, −3.62152119197582334243797673319, −2.56485909336390661346747036875, −1.97429829735326178112633537233, −0.73689445958994060240958765981,
0.73689445958994060240958765981, 1.97429829735326178112633537233, 2.56485909336390661346747036875, 3.62152119197582334243797673319, 4.23458362790248218519386092577, 4.88562457715919204178376484130, 5.85588904112827243672923891660, 6.52995576064869421005831147608, 7.28517638205992651476170625719, 7.994920884572981285922036248461