| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 2·13-s − 15-s + 6·17-s − 2·19-s − 4·21-s + 23-s + 25-s − 27-s + 6·29-s + 4·31-s + 4·35-s + 8·37-s − 2·39-s + 6·41-s − 8·43-s + 45-s − 12·47-s + 9·49-s − 6·51-s − 6·53-s + 2·57-s + 6·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 1.45·17-s − 0.458·19-s − 0.872·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.676·35-s + 1.31·37-s − 0.320·39-s + 0.937·41-s − 1.21·43-s + 0.149·45-s − 1.75·47-s + 9/7·49-s − 0.840·51-s − 0.824·53-s + 0.264·57-s + 0.781·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.532549003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.532549003\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.985258276404753820262702007381, −7.69137159697916677212527046147, −6.46308529675286474850330381553, −6.10637523571841171749990073885, −5.01626410668179427795341023017, −4.87189977067826181898334026335, −3.80189606860667529215445769860, −2.73015648721643364351441045002, −1.61806562620190443301969498989, −0.986029338559047467037060225637,
0.986029338559047467037060225637, 1.61806562620190443301969498989, 2.73015648721643364351441045002, 3.80189606860667529215445769860, 4.87189977067826181898334026335, 5.01626410668179427795341023017, 6.10637523571841171749990073885, 6.46308529675286474850330381553, 7.69137159697916677212527046147, 7.985258276404753820262702007381
