| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 3·9-s − 2·10-s − 11-s + 2·13-s + 16-s + 2·17-s + 3·18-s + 4·19-s + 2·20-s + 22-s + 8·23-s − 25-s − 2·26-s + 2·29-s − 32-s − 2·34-s − 3·36-s + 2·37-s − 4·38-s − 2·40-s + 10·41-s + 43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 9-s − 0.632·10-s − 0.301·11-s + 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.707·18-s + 0.917·19-s + 0.447·20-s + 0.213·22-s + 1.66·23-s − 1/5·25-s − 0.392·26-s + 0.371·29-s − 0.176·32-s − 0.342·34-s − 1/2·36-s + 0.328·37-s − 0.648·38-s − 0.316·40-s + 1.56·41-s + 0.152·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.309185292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.309185292\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.779121454543758695462614347263, −9.361163745386668069674032411183, −8.455880728734011695353593452492, −7.69813405024567632907308087150, −6.61708909684839374605798248991, −5.80468499116263203783036886126, −5.08581686777542773746204593488, −3.35213814772372799252657670682, −2.45556347160025075437960985248, −1.04809938414014428546230540055,
1.04809938414014428546230540055, 2.45556347160025075437960985248, 3.35213814772372799252657670682, 5.08581686777542773746204593488, 5.80468499116263203783036886126, 6.61708909684839374605798248991, 7.69813405024567632907308087150, 8.455880728734011695353593452492, 9.361163745386668069674032411183, 9.779121454543758695462614347263
