L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 6·11-s + 5·13-s + 14-s + 16-s + 6·17-s + 2·19-s + 6·22-s + 5·26-s + 28-s + 9·29-s + 5·31-s + 32-s + 6·34-s − 4·37-s + 2·38-s − 3·41-s − 4·43-s + 6·44-s + 6·47-s + 49-s + 5·52-s − 9·53-s + 56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.80·11-s + 1.38·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.458·19-s + 1.27·22-s + 0.980·26-s + 0.188·28-s + 1.67·29-s + 0.898·31-s + 0.176·32-s + 1.02·34-s − 0.657·37-s + 0.324·38-s − 0.468·41-s − 0.609·43-s + 0.904·44-s + 0.875·47-s + 1/7·49-s + 0.693·52-s − 1.23·53-s + 0.133·56-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
\(L(1)\) |
\(\approx\) |
\(5.128066017\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.128066017\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−7.61866683013096963358069336638, −6.80783983865841558751810545132, −6.25198129397399697270855845990, −5.76129952039582737876112227741, −4.82601575864466823345176365956, −4.20121099041644421536914061825, −3.47105463659034013747838819196, −2.94300014488005244548785112028, −1.40516340681673494090259273280, −1.24952410818763627748009255385,
1.24952410818763627748009255385, 1.40516340681673494090259273280, 2.94300014488005244548785112028, 3.47105463659034013747838819196, 4.20121099041644421536914061825, 4.82601575864466823345176365956, 5.76129952039582737876112227741, 6.25198129397399697270855845990, 6.80783983865841558751810545132, 7.61866683013096963358069336638