| L(s) = 1 | + 3·3-s + 7-s + 6·9-s − 4·13-s + 19-s + 3·21-s − 6·23-s + 9·27-s − 3·29-s + 9·31-s + 6·37-s − 12·39-s + 6·41-s − 6·43-s + 47-s + 49-s − 13·53-s + 3·57-s + 7·59-s + 10·61-s + 6·63-s − 6·67-s − 18·69-s − 6·71-s + 2·73-s − 16·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.377·7-s + 2·9-s − 1.10·13-s + 0.229·19-s + 0.654·21-s − 1.25·23-s + 1.73·27-s − 0.557·29-s + 1.61·31-s + 0.986·37-s − 1.92·39-s + 0.937·41-s − 0.914·43-s + 0.145·47-s + 1/7·49-s − 1.78·53-s + 0.397·57-s + 0.911·59-s + 1.28·61-s + 0.755·63-s − 0.733·67-s − 2.16·69-s − 0.712·71-s + 0.234·73-s − 1.80·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.25139750170423, −13.05571770754872, −12.41539177895965, −11.98314693153914, −11.46516056776543, −10.94588290938849, −10.09058681746172, −9.859448302721742, −9.655336852770740, −8.977654257359515, −8.397615715539814, −8.158313773831957, −7.711139209600658, −7.211117909811562, −6.823233492558137, −6.011611617021533, −5.548525915141465, −4.628530268365566, −4.432347408902818, −3.903750540222979, −3.125190859424647, −2.784709776741059, −2.237883356625605, −1.733740761384817, −1.046948822587578, 0,
1.046948822587578, 1.733740761384817, 2.237883356625605, 2.784709776741059, 3.125190859424647, 3.903750540222979, 4.432347408902818, 4.628530268365566, 5.548525915141465, 6.011611617021533, 6.823233492558137, 7.211117909811562, 7.711139209600658, 8.158313773831957, 8.397615715539814, 8.977654257359515, 9.655336852770740, 9.859448302721742, 10.09058681746172, 10.94588290938849, 11.46516056776543, 11.98314693153914, 12.41539177895965, 13.05571770754872, 13.25139750170423
