L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 5·11-s + 5·13-s + 14-s + 16-s − 3·17-s − 6·19-s + 5·22-s − 23-s + 5·26-s + 28-s + 29-s + 31-s + 32-s − 3·34-s + 10·37-s − 6·38-s + 4·41-s − 5·43-s + 5·44-s − 46-s + 11·47-s + 49-s + 5·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.50·11-s + 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 1.37·19-s + 1.06·22-s − 0.208·23-s + 0.980·26-s + 0.188·28-s + 0.185·29-s + 0.179·31-s + 0.176·32-s − 0.514·34-s + 1.64·37-s − 0.973·38-s + 0.624·41-s − 0.762·43-s + 0.753·44-s − 0.147·46-s + 1.60·47-s + 1/7·49-s + 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.299820650\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.299820650\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.60367659276136143532683787410, −6.75708871821180913671349610126, −6.23043605356616924172306335379, −5.87767750840003424061308994423, −4.71670048391958710231770758376, −4.12462750527423702764999188842, −3.76859968161957839199952640915, −2.64034339280279353444967375718, −1.79681707436529722748658006736, −0.946937733385361323336215895109,
0.946937733385361323336215895109, 1.79681707436529722748658006736, 2.64034339280279353444967375718, 3.76859968161957839199952640915, 4.12462750527423702764999188842, 4.71670048391958710231770758376, 5.87767750840003424061308994423, 6.23043605356616924172306335379, 6.75708871821180913671349610126, 7.60367659276136143532683787410