L(s) = 1 | − 3-s − 5-s + 9-s − 6·11-s − 4·13-s + 15-s − 6·17-s + 6·19-s + 4·23-s + 25-s − 27-s − 2·29-s − 2·31-s + 6·33-s + 6·37-s + 4·39-s − 6·41-s + 12·43-s − 45-s + 6·51-s − 4·53-s + 6·55-s − 6·57-s − 4·59-s + 10·61-s + 4·65-s + 12·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.80·11-s − 1.10·13-s + 0.258·15-s − 1.45·17-s + 1.37·19-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.359·31-s + 1.04·33-s + 0.986·37-s + 0.640·39-s − 0.937·41-s + 1.82·43-s − 0.149·45-s + 0.840·51-s − 0.549·53-s + 0.809·55-s − 0.794·57-s − 0.520·59-s + 1.28·61-s + 0.496·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−15.61396556713147, −15.51142156419772, −14.68709798609492, −14.15850839704161, −13.34062188168895, −12.91599213518486, −12.66662767330566, −11.74633135376231, −11.42426141566537, −10.83681165671731, −10.37232193475766, −9.701762447351764, −9.208875251106237, −8.416988881376152, −7.771164396081124, −7.241906381212852, −6.969088108882599, −5.879854949813932, −5.438915485328004, −4.756992065941002, −4.448129274732225, −3.331724936782141, −2.688394519765844, −2.061494212271190, −0.7738017414448675, 0,
0.7738017414448675, 2.061494212271190, 2.688394519765844, 3.331724936782141, 4.448129274732225, 4.756992065941002, 5.438915485328004, 5.879854949813932, 6.969088108882599, 7.241906381212852, 7.771164396081124, 8.416988881376152, 9.208875251106237, 9.701762447351764, 10.37232193475766, 10.83681165671731, 11.42426141566537, 11.74633135376231, 12.66662767330566, 12.91599213518486, 13.34062188168895, 14.15850839704161, 14.68709798609492, 15.51142156419772, 15.61396556713147