L(s) = 1 | + 3-s − 5-s + 9-s − 6·13-s − 15-s + 2·17-s − 6·19-s + 23-s + 25-s + 27-s + 6·29-s + 4·31-s + 4·37-s − 6·39-s + 6·41-s − 4·43-s − 45-s + 4·47-s − 7·49-s + 2·51-s + 6·53-s − 6·57-s − 14·59-s + 6·61-s + 6·65-s + 4·67-s + 69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.66·13-s − 0.258·15-s + 0.485·17-s − 1.37·19-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.657·37-s − 0.960·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 0.583·47-s − 49-s + 0.280·51-s + 0.824·53-s − 0.794·57-s − 1.82·59-s + 0.768·61-s + 0.744·65-s + 0.488·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.867814387\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.867814387\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.009109240003099559482733202672, −7.65259866236025660798870647766, −6.79315804538354942809382444641, −6.18291015017550627584684168714, −4.95405103454609140206955149268, −4.57687473547879907191422494813, −3.64832798509651947374533011926, −2.73571240150927353624122152013, −2.12043241937554226688175790414, −0.69079739527451739617645826986,
0.69079739527451739617645826986, 2.12043241937554226688175790414, 2.73571240150927353624122152013, 3.64832798509651947374533011926, 4.57687473547879907191422494813, 4.95405103454609140206955149268, 6.18291015017550627584684168714, 6.79315804538354942809382444641, 7.65259866236025660798870647766, 8.009109240003099559482733202672