L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 5·11-s + 5·13-s + 15-s + 17-s − 5·19-s + 21-s + 23-s − 4·25-s − 27-s + 6·29-s + 6·31-s + 5·33-s + 35-s − 4·37-s − 5·39-s + 7·41-s − 7·43-s − 45-s − 6·47-s + 49-s − 51-s − 6·53-s + 5·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 1.38·13-s + 0.258·15-s + 0.242·17-s − 1.14·19-s + 0.218·21-s + 0.208·23-s − 4/5·25-s − 0.192·27-s + 1.11·29-s + 1.07·31-s + 0.870·33-s + 0.169·35-s − 0.657·37-s − 0.800·39-s + 1.09·41-s − 1.06·43-s − 0.149·45-s − 0.875·47-s + 1/7·49-s − 0.140·51-s − 0.824·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.80857506947034, −15.46979699711997, −14.82343487084406, −14.04505955645753, −13.37816410545009, −13.09382130831707, −12.58734233104592, −11.83415972623794, −11.47531781180809, −10.70492627878955, −10.41859327474591, −9.941108168600991, −9.047029748643226, −8.327126755465634, −8.081722660350342, −7.358389230491419, −6.470997388356633, −6.254346949191078, −5.472446709052511, −4.838591779949009, −4.198589283601330, −3.470243095012436, −2.795840713946795, −1.911363528150883, −0.8627079721000353, 0,
0.8627079721000353, 1.911363528150883, 2.795840713946795, 3.470243095012436, 4.198589283601330, 4.838591779949009, 5.472446709052511, 6.254346949191078, 6.470997388356633, 7.358389230491419, 8.081722660350342, 8.327126755465634, 9.047029748643226, 9.941108168600991, 10.41859327474591, 10.70492627878955, 11.47531781180809, 11.83415972623794, 12.58734233104592, 13.09382130831707, 13.37816410545009, 14.04505955645753, 14.82343487084406, 15.46979699711997, 15.80857506947034