L(s) = 1 | + 5-s − 2·7-s − 4·11-s − 13-s + 8·17-s + 2·19-s − 4·23-s + 25-s − 8·29-s − 10·31-s − 2·35-s + 6·37-s + 6·41-s + 8·43-s + 8·47-s − 3·49-s − 12·53-s − 4·55-s + 4·59-s + 10·61-s − 65-s − 2·67-s + 6·73-s + 8·77-s − 12·79-s + 4·83-s + 8·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.20·11-s − 0.277·13-s + 1.94·17-s + 0.458·19-s − 0.834·23-s + 1/5·25-s − 1.48·29-s − 1.79·31-s − 0.338·35-s + 0.986·37-s + 0.937·41-s + 1.21·43-s + 1.16·47-s − 3/7·49-s − 1.64·53-s − 0.539·55-s + 0.520·59-s + 1.28·61-s − 0.124·65-s − 0.244·67-s + 0.702·73-s + 0.911·77-s − 1.35·79-s + 0.439·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.582757514\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.582757514\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.48647681393247181697149130450, −7.36099562505490966252278695617, −6.08720846018516294588661755991, −5.66365274996790647162399011237, −5.23954046198580642777490498223, −4.08999588039270914326447322955, −3.36312345080183111601665529508, −2.67524847862010056627725919027, −1.80874015634772738613211189005, −0.58769404101495007216785436587,
0.58769404101495007216785436587, 1.80874015634772738613211189005, 2.67524847862010056627725919027, 3.36312345080183111601665529508, 4.08999588039270914326447322955, 5.23954046198580642777490498223, 5.66365274996790647162399011237, 6.08720846018516294588661755991, 7.36099562505490966252278695617, 7.48647681393247181697149130450