| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s − 13-s + 2·15-s + 17-s − 4·19-s + 21-s − 4·23-s − 25-s − 27-s + 5·29-s + 7·31-s + 2·35-s + 9·37-s + 39-s − 8·41-s − 7·43-s − 2·45-s + 7·47-s − 6·49-s − 51-s − 4·53-s + 4·57-s + 7·59-s + 14·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.516·15-s + 0.242·17-s − 0.917·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.928·29-s + 1.25·31-s + 0.338·35-s + 1.47·37-s + 0.160·39-s − 1.24·41-s − 1.06·43-s − 0.298·45-s + 1.02·47-s − 6/7·49-s − 0.140·51-s − 0.549·53-s + 0.529·57-s + 0.911·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.392816364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.392816364\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.56503923266172, −12.01988934747962, −11.73889742399802, −11.46259229589040, −10.82016138774006, −10.32925536093423, −9.914096218757320, −9.671195502223396, −8.861730424467769, −8.339837656261442, −8.030400988367099, −7.634967721879849, −6.890295894750609, −6.562167229237568, −6.186755329442824, −5.613368128642556, −4.840823231725882, −4.693703482429881, −3.923188289012181, −3.689012320048927, −2.953016962593401, −2.339283635881637, −1.767025694480862, −0.7875332566447591, −0.4442452632019937,
0.4442452632019937, 0.7875332566447591, 1.767025694480862, 2.339283635881637, 2.953016962593401, 3.689012320048927, 3.923188289012181, 4.693703482429881, 4.840823231725882, 5.613368128642556, 6.186755329442824, 6.562167229237568, 6.890295894750609, 7.634967721879849, 8.030400988367099, 8.339837656261442, 8.861730424467769, 9.671195502223396, 9.914096218757320, 10.32925536093423, 10.82016138774006, 11.46259229589040, 11.73889742399802, 12.01988934747962, 12.56503923266172
