L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 11-s − 2·13-s + 15-s + 2·17-s − 4·19-s − 21-s − 4·23-s + 25-s + 27-s + 6·29-s + 33-s − 35-s − 2·37-s − 2·39-s + 6·41-s − 12·43-s + 45-s + 8·47-s + 49-s + 2·51-s + 6·53-s + 55-s − 4·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.174·33-s − 0.169·35-s − 0.328·37-s − 0.320·39-s + 0.937·41-s − 1.82·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 0.134·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.881239165\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.881239165\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.18920827823446, −13.62640511664024, −13.12057417054423, −12.63941654337572, −12.11575010551695, −11.78372040727661, −10.96160291884111, −10.36816654121808, −10.01801185222730, −9.638565396671604, −8.900099141334052, −8.614283281849621, −7.980912790172628, −7.432719983013342, −6.824474549507201, −6.343905466741431, −5.842074091800760, −5.100491895993070, −4.557618254447874, −3.882144763417925, −3.383552059411183, −2.567767054528919, −2.203958254564000, −1.407208683094023, −0.5356816459293311,
0.5356816459293311, 1.407208683094023, 2.203958254564000, 2.567767054528919, 3.383552059411183, 3.882144763417925, 4.557618254447874, 5.100491895993070, 5.842074091800760, 6.343905466741431, 6.824474549507201, 7.432719983013342, 7.980912790172628, 8.614283281849621, 8.900099141334052, 9.638565396671604, 10.01801185222730, 10.36816654121808, 10.96160291884111, 11.78372040727661, 12.11575010551695, 12.63941654337572, 13.12057417054423, 13.62640511664024, 14.18920827823446