| L(s) = 1 | + 2-s + 4-s + 4.69·7-s + 8-s − 6.40·11-s − 1.06·13-s + 4.69·14-s + 16-s + 1.91·17-s − 19-s − 6.40·22-s + 1.79·23-s − 1.06·26-s + 4.69·28-s − 2.93·29-s − 5.55·31-s + 32-s + 1.91·34-s + 11.4·37-s − 38-s + 1.14·41-s + 3.55·43-s − 6.40·44-s + 1.79·46-s + 10.8·47-s + 15.0·49-s − 1.06·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.77·7-s + 0.353·8-s − 1.93·11-s − 0.295·13-s + 1.25·14-s + 0.250·16-s + 0.465·17-s − 0.229·19-s − 1.36·22-s + 0.374·23-s − 0.208·26-s + 0.887·28-s − 0.545·29-s − 0.997·31-s + 0.176·32-s + 0.328·34-s + 1.87·37-s − 0.162·38-s + 0.178·41-s + 0.542·43-s − 0.966·44-s + 0.264·46-s + 1.58·47-s + 2.14·49-s − 0.147·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.795571463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.795571463\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 4.69T + 7T^{2} \) |
| 11 | \( 1 + 6.40T + 11T^{2} \) |
| 13 | \( 1 + 1.06T + 13T^{2} \) |
| 17 | \( 1 - 1.91T + 17T^{2} \) |
| 23 | \( 1 - 1.79T + 23T^{2} \) |
| 29 | \( 1 + 2.93T + 29T^{2} \) |
| 31 | \( 1 + 5.55T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 - 1.14T + 41T^{2} \) |
| 43 | \( 1 - 3.55T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 8.69T + 53T^{2} \) |
| 59 | \( 1 - 5.63T + 59T^{2} \) |
| 61 | \( 1 + 3.39T + 61T^{2} \) |
| 67 | \( 1 - 8.82T + 67T^{2} \) |
| 71 | \( 1 - 1.42T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 1.96T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−7.62533134698806928934619752650, −7.44210278199316324917238706120, −6.16848512766407757022816162584, −5.43550523823473992024385017535, −5.04740447966574022547986681665, −4.47884954130528751319258693914, −3.56655545436772841553862821170, −2.46843252328081942670239562208, −2.10658192690874479963155257130, −0.855546094780470228366999836326,
0.855546094780470228366999836326, 2.10658192690874479963155257130, 2.46843252328081942670239562208, 3.56655545436772841553862821170, 4.47884954130528751319258693914, 5.04740447966574022547986681665, 5.43550523823473992024385017535, 6.16848512766407757022816162584, 7.44210278199316324917238706120, 7.62533134698806928934619752650
