L(s) = 1 | + 3-s − 5-s + 3.56·7-s + 9-s − 2·11-s − 2·13-s − 15-s + 1.56·17-s − 7.12·19-s + 3.56·21-s + 23-s + 25-s + 27-s + 0.438·29-s − 8.68·31-s − 2·33-s − 3.56·35-s − 8.68·37-s − 2·39-s − 6.68·41-s − 45-s − 6.24·47-s + 5.68·49-s + 1.56·51-s − 7.56·53-s + 2·55-s − 7.12·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.34·7-s + 0.333·9-s − 0.603·11-s − 0.554·13-s − 0.258·15-s + 0.378·17-s − 1.63·19-s + 0.777·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.0814·29-s − 1.55·31-s − 0.348·33-s − 0.602·35-s − 1.42·37-s − 0.320·39-s − 1.04·41-s − 0.149·45-s − 0.911·47-s + 0.812·49-s + 0.218·51-s − 1.03·53-s + 0.269·55-s − 0.943·57-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 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 1.56T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 29 | \( 1 - 0.438T + 29T^{2} \) |
| 31 | \( 1 + 8.68T + 31T^{2} \) |
| 37 | \( 1 + 8.68T + 37T^{2} \) |
| 41 | \( 1 + 6.68T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 + 7.56T + 53T^{2} \) |
| 59 | \( 1 - 1.56T + 59T^{2} \) |
| 61 | \( 1 - 2.24T + 61T^{2} \) |
| 67 | \( 1 - 0.684T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 6.68T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−7.969952906773443476712744861884, −7.25657399021338511070295133645, −6.56664502989478303509099222724, −5.33355993157539887092231530642, −4.91792692654537727859290519160, −4.09405235038587145134888758057, −3.29696159947264985592680675113, −2.22411115314445032238417388992, −1.61086890532997028216013376416, 0,
1.61086890532997028216013376416, 2.22411115314445032238417388992, 3.29696159947264985592680675113, 4.09405235038587145134888758057, 4.91792692654537727859290519160, 5.33355993157539887092231530642, 6.56664502989478303509099222724, 7.25657399021338511070295133645, 7.969952906773443476712744861884