L(s) = 1 | + 0.656·3-s − 2.09·5-s − 2.56·9-s − 1.84·11-s + 2.10·13-s − 1.37·15-s − 4.60·17-s − 8.45·19-s + 23-s − 0.625·25-s − 3.65·27-s − 2.24·29-s + 10.1·31-s − 1.20·33-s − 6.27·37-s + 1.38·39-s − 0.725·41-s − 10.8·43-s + 5.37·45-s + 2.14·47-s − 3.02·51-s + 12.8·53-s + 3.85·55-s − 5.54·57-s − 1.37·59-s + 1.80·61-s − 4.40·65-s + ⋯ |
L(s) = 1 | + 0.378·3-s − 0.935·5-s − 0.856·9-s − 0.555·11-s + 0.584·13-s − 0.354·15-s − 1.11·17-s − 1.93·19-s + 0.208·23-s − 0.125·25-s − 0.703·27-s − 0.417·29-s + 1.81·31-s − 0.210·33-s − 1.03·37-s + 0.221·39-s − 0.113·41-s − 1.64·43-s + 0.801·45-s + 0.313·47-s − 0.423·51-s + 1.76·53-s + 0.519·55-s − 0.734·57-s − 0.179·59-s + 0.230·61-s − 0.546·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7820907762\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7820907762\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 0.656T + 3T^{2} \) |
| 5 | \( 1 + 2.09T + 5T^{2} \) |
| 11 | \( 1 + 1.84T + 11T^{2} \) |
| 13 | \( 1 - 2.10T + 13T^{2} \) |
| 17 | \( 1 + 4.60T + 17T^{2} \) |
| 19 | \( 1 + 8.45T + 19T^{2} \) |
| 29 | \( 1 + 2.24T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 6.27T + 37T^{2} \) |
| 41 | \( 1 + 0.725T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 2.14T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 1.37T + 59T^{2} \) |
| 61 | \( 1 - 1.80T + 61T^{2} \) |
| 67 | \( 1 + 5.83T + 67T^{2} \) |
| 71 | \( 1 - 4.43T + 71T^{2} \) |
| 73 | \( 1 - 4.08T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 + 1.78T + 83T^{2} \) |
| 89 | \( 1 + 3.69T + 89T^{2} \) |
| 97 | \( 1 - 3.34T + 97T^{2} \) |
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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
−8.010365840185012907855376142566, −7.00538858794805320134123262372, −6.48800069486855154455163316095, −5.72444093246485048707646221736, −4.81566837179426739943103463255, −4.15143668334990567298691652883, −3.50947989694774190437296727007, −2.63767652963890103500937553785, −1.94814040350850783146984090691, −0.39299692369410535603584922309,
0.39299692369410535603584922309, 1.94814040350850783146984090691, 2.63767652963890103500937553785, 3.50947989694774190437296727007, 4.15143668334990567298691652883, 4.81566837179426739943103463255, 5.72444093246485048707646221736, 6.48800069486855154455163316095, 7.00538858794805320134123262372, 8.010365840185012907855376142566