| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s − 4·11-s + 13-s + 15-s − 6·17-s − 4·21-s + 8·23-s + 25-s − 27-s + 6·29-s + 4·31-s + 4·33-s − 4·35-s − 2·37-s − 39-s − 10·41-s + 4·43-s − 45-s + 8·47-s + 9·49-s + 6·51-s − 2·53-s + 4·55-s − 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 0.258·15-s − 1.45·17-s − 0.872·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.696·33-s − 0.676·35-s − 0.328·37-s − 0.160·39-s − 1.56·41-s + 0.609·43-s − 0.149·45-s + 1.16·47-s + 9/7·49-s + 0.840·51-s − 0.274·53-s + 0.539·55-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.469993090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.469993090\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.453727661169848939735174320956, −8.072756676496475639971313161278, −7.17516968512342369948876300218, −6.54658424737806712617173050247, −5.38010853571471058595867691512, −4.85719285905544219284711770884, −4.33726130108018425784375424189, −2.99221031007625175900190168785, −1.97927761175828111564529437375, −0.76404989359937815787458235604,
0.76404989359937815787458235604, 1.97927761175828111564529437375, 2.99221031007625175900190168785, 4.33726130108018425784375424189, 4.85719285905544219284711770884, 5.38010853571471058595867691512, 6.54658424737806712617173050247, 7.17516968512342369948876300218, 8.072756676496475639971313161278, 8.453727661169848939735174320956
