L(s) = 1 | + 7-s + 2·11-s + 2·17-s − 5·25-s + 6·29-s − 4·31-s + 2·37-s + 12·43-s + 2·47-s + 49-s − 6·53-s − 6·59-s − 14·61-s + 12·67-s − 10·71-s + 10·73-s + 2·77-s − 2·83-s − 4·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 2·119-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 0.603·11-s + 0.485·17-s − 25-s + 1.11·29-s − 0.718·31-s + 0.328·37-s + 1.82·43-s + 0.291·47-s + 1/7·49-s − 0.824·53-s − 0.781·59-s − 1.79·61-s + 1.46·67-s − 1.18·71-s + 1.17·73-s + 0.227·77-s − 0.219·83-s − 0.423·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.21293859440197, −13.80933443529452, −13.22227215426477, −12.56582161213961, −12.18821191076570, −11.80749833167526, −11.09400468153357, −10.83934501294155, −10.18611068222417, −9.616404150547915, −9.188361114581150, −8.718152534993989, −7.965364445955816, −7.686402030784983, −7.137320404721428, −6.297619533274103, −6.115020020311570, −5.341685646009655, −4.834784321149529, −4.121103951822117, −3.773675783257445, −2.939257772820024, −2.372126794994414, −1.547498965782213, −1.025532912705422, 0,
1.025532912705422, 1.547498965782213, 2.372126794994414, 2.939257772820024, 3.773675783257445, 4.121103951822117, 4.834784321149529, 5.341685646009655, 6.115020020311570, 6.297619533274103, 7.137320404721428, 7.686402030784983, 7.965364445955816, 8.718152534993989, 9.188361114581150, 9.616404150547915, 10.18611068222417, 10.83934501294155, 11.09400468153357, 11.80749833167526, 12.18821191076570, 12.56582161213961, 13.22227215426477, 13.80933443529452, 14.21293859440197