| L(s) = 1 | + 2·3-s + 7-s + 9-s + 3·11-s + 5·13-s + 6·17-s + 7·19-s + 2·21-s + 23-s − 4·27-s + 3·29-s − 2·31-s + 6·33-s + 8·37-s + 10·39-s − 9·41-s + 7·43-s − 6·47-s − 6·49-s + 12·51-s − 12·53-s + 14·57-s + 8·61-s + 63-s + 4·67-s + 2·69-s − 12·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.38·13-s + 1.45·17-s + 1.60·19-s + 0.436·21-s + 0.208·23-s − 0.769·27-s + 0.557·29-s − 0.359·31-s + 1.04·33-s + 1.31·37-s + 1.60·39-s − 1.40·41-s + 1.06·43-s − 0.875·47-s − 6/7·49-s + 1.68·51-s − 1.64·53-s + 1.85·57-s + 1.02·61-s + 0.125·63-s + 0.488·67-s + 0.240·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.446556897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.446556897\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−7.937147856423671593469758586913, −7.24353661293005764774496738799, −6.34969186414046345862165529456, −5.71221269968579547222757284407, −4.92143487512423462377077855591, −3.91510281260016986534796411120, −3.37354510829483043088402509287, −2.88051877115012643201064159769, −1.59593217578268861216226726359, −1.10633221179109808238469054832,
1.10633221179109808238469054832, 1.59593217578268861216226726359, 2.88051877115012643201064159769, 3.37354510829483043088402509287, 3.91510281260016986534796411120, 4.92143487512423462377077855591, 5.71221269968579547222757284407, 6.34969186414046345862165529456, 7.24353661293005764774496738799, 7.937147856423671593469758586913
