L(s) = 1 | + 3-s − 5-s + 9-s − 2·11-s − 2·13-s − 15-s − 4·17-s + 8·23-s + 25-s + 27-s − 2·31-s − 2·33-s − 8·37-s − 2·39-s − 2·41-s + 2·43-s − 45-s + 10·47-s − 4·51-s + 2·53-s + 2·55-s − 4·59-s + 10·61-s + 2·65-s − 2·67-s + 8·69-s − 12·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.258·15-s − 0.970·17-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.359·31-s − 0.348·33-s − 1.31·37-s − 0.320·39-s − 0.312·41-s + 0.304·43-s − 0.149·45-s + 1.45·47-s − 0.560·51-s + 0.274·53-s + 0.269·55-s − 0.520·59-s + 1.28·61-s + 0.248·65-s − 0.244·67-s + 0.963·69-s − 1.42·71-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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.90540874726970, −14.37444148143305, −13.84346198789049, −13.14563246527824, −13.03228544199373, −12.27886750525504, −11.84535935757682, −11.17558132042255, −10.51749834895715, −10.44869148073195, −9.375496386128104, −9.121675795885057, −8.588911488395000, −7.984123664080509, −7.418406421313188, −6.960044129449325, −6.511889510499270, −5.448944501070151, −5.114725907710240, −4.412851398744388, −3.831744814132777, −3.105010915384902, −2.568660967040397, −1.927909604905227, −0.9380179014977944, 0,
0.9380179014977944, 1.927909604905227, 2.568660967040397, 3.105010915384902, 3.831744814132777, 4.412851398744388, 5.114725907710240, 5.448944501070151, 6.511889510499270, 6.960044129449325, 7.418406421313188, 7.984123664080509, 8.588911488395000, 9.121675795885057, 9.375496386128104, 10.44869148073195, 10.51749834895715, 11.17558132042255, 11.84535935757682, 12.27886750525504, 13.03228544199373, 13.14563246527824, 13.84346198789049, 14.37444148143305, 14.90540874726970