L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 6.24·11-s − 2.58·13-s − 15-s − 1.58·17-s − 7.07·19-s − 21-s + 23-s + 25-s − 27-s − 7.24·29-s − 0.656·31-s − 6.24·33-s + 35-s − 3·37-s + 2.58·39-s − 4.41·41-s − 2·43-s + 45-s − 2.58·47-s − 6·49-s + 1.58·51-s − 3.58·53-s + 6.24·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s + 1.88·11-s − 0.717·13-s − 0.258·15-s − 0.384·17-s − 1.62·19-s − 0.218·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s − 1.34·29-s − 0.117·31-s − 1.08·33-s + 0.169·35-s − 0.493·37-s + 0.414·39-s − 0.689·41-s − 0.304·43-s + 0.149·45-s − 0.377·47-s − 0.857·49-s + 0.222·51-s − 0.492·53-s + 0.841·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 6.24T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 + 1.58T + 17T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 29 | \( 1 + 7.24T + 29T^{2} \) |
| 31 | \( 1 + 0.656T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + 4.41T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 + 3.58T + 53T^{2} \) |
| 59 | \( 1 + 2.07T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 0.656T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 + 1.92T + 83T^{2} \) |
| 89 | \( 1 + 5.17T + 89T^{2} \) |
| 97 | \( 1 + 6.82T + 97T^{2} \) |
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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
−7.71143626023053175143171229734, −6.81829885701393525496777589447, −6.47135976633997813671930328080, −5.73200606517624761887001746110, −4.80062121280810881805876208081, −4.27045187131788394301113708874, −3.39035498720159624891161968549, −2.03660463170726851993443952445, −1.48798733054522071923383153400, 0,
1.48798733054522071923383153400, 2.03660463170726851993443952445, 3.39035498720159624891161968549, 4.27045187131788394301113708874, 4.80062121280810881805876208081, 5.73200606517624761887001746110, 6.47135976633997813671930328080, 6.81829885701393525496777589447, 7.71143626023053175143171229734