L(s) = 1 | − 3-s + 7-s + 9-s − 2·11-s + 6·13-s + 4·17-s + 6·19-s − 21-s + 8·23-s − 27-s + 6·29-s + 2·31-s + 2·33-s + 4·37-s − 6·39-s + 2·41-s − 4·43-s − 8·47-s + 49-s − 4·51-s + 6·53-s − 6·57-s + 8·59-s − 10·61-s + 63-s − 8·67-s − 8·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.66·13-s + 0.970·17-s + 1.37·19-s − 0.218·21-s + 1.66·23-s − 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.348·33-s + 0.657·37-s − 0.960·39-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.560·51-s + 0.824·53-s − 0.794·57-s + 1.04·59-s − 1.28·61-s + 0.125·63-s − 0.977·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.313612330\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.313612330\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.84167180600735126849465819116, −7.05693507279556783124275735264, −6.38967288915241187005687210081, −5.60673339084637943352854710400, −5.16581457150367048338818862604, −4.39180611473697080382301879227, −3.39416021405628080146605186483, −2.84713084030657961161040845498, −1.38360896712049166556700674056, −0.901580883063587347430228827389,
0.901580883063587347430228827389, 1.38360896712049166556700674056, 2.84713084030657961161040845498, 3.39416021405628080146605186483, 4.39180611473697080382301879227, 5.16581457150367048338818862604, 5.60673339084637943352854710400, 6.38967288915241187005687210081, 7.05693507279556783124275735264, 7.84167180600735126849465819116