L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 2·11-s + 4·13-s + 15-s − 6·17-s + 4·19-s + 21-s + 23-s + 25-s − 27-s + 2·29-s − 2·31-s − 2·33-s + 35-s + 4·37-s − 4·39-s − 2·41-s − 4·43-s − 45-s + 4·47-s + 49-s + 6·51-s − 12·53-s − 2·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 0.258·15-s − 1.45·17-s + 0.917·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.359·31-s − 0.348·33-s + 0.169·35-s + 0.657·37-s − 0.640·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s + 0.583·47-s + 1/7·49-s + 0.840·51-s − 1.64·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.20668150417181, −14.67146757046981, −13.83682864385849, −13.55356240123667, −13.04310118395523, −12.31696858503635, −12.02788237616181, −11.22580321332223, −11.08272719610926, −10.55020849007539, −9.724053968924797, −9.215923484986202, −8.828794018423167, −8.063237199621792, −7.556645549444277, −6.773123909409317, −6.413128757507382, −5.991913242030464, −5.059034120949196, −4.648094063805812, −3.838627053793286, −3.463879764179705, −2.596142049828881, −1.634561170824963, −0.9334099865617281, 0,
0.9334099865617281, 1.634561170824963, 2.596142049828881, 3.463879764179705, 3.838627053793286, 4.648094063805812, 5.059034120949196, 5.991913242030464, 6.413128757507382, 6.773123909409317, 7.556645549444277, 8.063237199621792, 8.828794018423167, 9.215923484986202, 9.724053968924797, 10.55020849007539, 11.08272719610926, 11.22580321332223, 12.02788237616181, 12.31696858503635, 13.04310118395523, 13.55356240123667, 13.83682864385849, 14.67146757046981, 15.20668150417181