L(s) = 1 | + 3-s − 5-s + 9-s + 2·13-s − 15-s + 6·17-s − 4·19-s + 25-s + 27-s + 6·29-s + 4·31-s − 2·37-s + 2·39-s − 6·41-s − 8·43-s − 45-s + 12·47-s + 6·51-s − 6·53-s − 4·57-s − 12·59-s + 2·61-s − 2·65-s − 8·67-s − 14·73-s + 75-s − 16·79-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.328·37-s + 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.149·45-s + 1.75·47-s + 0.840·51-s − 0.824·53-s − 0.529·57-s − 1.56·59-s + 0.256·61-s − 0.248·65-s − 0.977·67-s − 1.63·73-s + 0.115·75-s − 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.04436674794512, −14.27208055695945, −13.85867760956367, −13.44052451244976, −12.77157133368668, −12.14774067095322, −12.01676513809551, −11.21701047717642, −10.58202602047985, −10.18436737873084, −9.726503758934634, −8.830915574212316, −8.609531122680396, −8.021255042064888, −7.548898318384794, −6.913226267826272, −6.305875378907583, −5.758351633271600, −4.956233909011682, −4.398766292466767, −3.795550673474404, −3.123175777548146, −2.724953823588603, −1.659367268736002, −1.123560417712240, 0,
1.123560417712240, 1.659367268736002, 2.724953823588603, 3.123175777548146, 3.795550673474404, 4.398766292466767, 4.956233909011682, 5.758351633271600, 6.305875378907583, 6.913226267826272, 7.548898318384794, 8.021255042064888, 8.609531122680396, 8.830915574212316, 9.726503758934634, 10.18436737873084, 10.58202602047985, 11.21701047717642, 12.01676513809551, 12.14774067095322, 12.77157133368668, 13.44052451244976, 13.85867760956367, 14.27208055695945, 15.04436674794512