L(s) = 1 | − 3·3-s + 5-s − 7-s + 6·9-s + 5·11-s + 3·13-s − 3·15-s − 17-s − 6·19-s + 3·21-s + 6·23-s + 25-s − 9·27-s + 9·29-s − 4·31-s − 15·33-s − 35-s − 2·37-s − 9·39-s − 4·41-s − 10·43-s + 6·45-s − 47-s + 49-s + 3·51-s − 4·53-s + 5·55-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 0.447·5-s − 0.377·7-s + 2·9-s + 1.50·11-s + 0.832·13-s − 0.774·15-s − 0.242·17-s − 1.37·19-s + 0.654·21-s + 1.25·23-s + 1/5·25-s − 1.73·27-s + 1.67·29-s − 0.718·31-s − 2.61·33-s − 0.169·35-s − 0.328·37-s − 1.44·39-s − 0.624·41-s − 1.52·43-s + 0.894·45-s − 0.145·47-s + 1/7·49-s + 0.420·51-s − 0.549·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.099912823\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.099912823\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.090841154864946897050602140573, −8.442630147086819400595758442541, −6.83120129452851121189770257485, −6.65835378060056758905376864628, −6.07952303847705355267793412469, −5.13650815214311653206052201867, −4.41082196705999392103931551577, −3.46157547207800553288566473698, −1.76847580206584462625237922454, −0.78311600712212551965990146664,
0.78311600712212551965990146664, 1.76847580206584462625237922454, 3.46157547207800553288566473698, 4.41082196705999392103931551577, 5.13650815214311653206052201867, 6.07952303847705355267793412469, 6.65835378060056758905376864628, 6.83120129452851121189770257485, 8.442630147086819400595758442541, 9.090841154864946897050602140573