| L(s) = 1 | + 3-s − 7-s + 9-s − 6·13-s + 2·17-s + 4·19-s − 21-s + 4·23-s + 27-s − 10·29-s − 8·31-s − 6·37-s − 6·39-s − 2·41-s + 4·43-s − 8·47-s + 49-s + 2·51-s + 10·53-s + 4·57-s + 12·59-s − 2·61-s − 63-s − 12·67-s + 4·69-s − 12·71-s + 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.66·13-s + 0.485·17-s + 0.917·19-s − 0.218·21-s + 0.834·23-s + 0.192·27-s − 1.85·29-s − 1.43·31-s − 0.986·37-s − 0.960·39-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.280·51-s + 1.37·53-s + 0.529·57-s + 1.56·59-s − 0.256·61-s − 0.125·63-s − 1.46·67-s + 0.481·69-s − 1.42·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.914571798209773752063091180197, −7.22705180533731249166256373485, −6.98516573751054829512564069856, −5.55277723147358999978364644764, −5.23189885714075765551668040516, −4.09370081306698782939745595865, −3.31500652764825935767069937194, −2.55481684789287360490968661545, −1.55522622494462511822699812767, 0,
1.55522622494462511822699812767, 2.55481684789287360490968661545, 3.31500652764825935767069937194, 4.09370081306698782939745595865, 5.23189885714075765551668040516, 5.55277723147358999978364644764, 6.98516573751054829512564069856, 7.22705180533731249166256373485, 7.914571798209773752063091180197
