L(s) = 1 | − 3·9-s + 4·11-s − 8·23-s − 5·25-s − 2·29-s + 6·37-s − 12·43-s + 10·53-s + 4·67-s − 16·71-s − 8·79-s + 9·81-s − 12·99-s − 20·107-s − 18·109-s + 2·113-s + ⋯ |
L(s) = 1 | − 9-s + 1.20·11-s − 1.66·23-s − 25-s − 0.371·29-s + 0.986·37-s − 1.82·43-s + 1.37·53-s + 0.488·67-s − 1.89·71-s − 0.900·79-s + 81-s − 1.20·99-s − 1.93·107-s − 1.72·109-s + 0.188·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.330759812635606547523055682566, −7.68998546052449448134866019451, −6.67625625855292974207346060772, −6.04800902507783081318745570011, −5.40928035513183831948820848756, −4.22989383620026669877623350668, −3.65330783308024527411814200780, −2.54910322548506501414462221460, −1.53416694723259152452558705643, 0,
1.53416694723259152452558705643, 2.54910322548506501414462221460, 3.65330783308024527411814200780, 4.22989383620026669877623350668, 5.40928035513183831948820848756, 6.04800902507783081318745570011, 6.67625625855292974207346060772, 7.68998546052449448134866019451, 8.330759812635606547523055682566