| L(s) = 1 | − 1.98·2-s + 0.183·3-s + 1.93·4-s − 1.97·5-s − 0.363·6-s − 5.05·7-s + 0.121·8-s − 2.96·9-s + 3.92·10-s − 1.03·11-s + 0.355·12-s − 2.23·13-s + 10.0·14-s − 0.362·15-s − 4.11·16-s + 3.42·17-s + 5.88·18-s + 19-s − 3.83·20-s − 0.925·21-s + 2.05·22-s + 7.05·23-s + 0.0223·24-s − 1.09·25-s + 4.44·26-s − 1.09·27-s − 9.79·28-s + ⋯ |
| L(s) = 1 | − 1.40·2-s + 0.105·3-s + 0.969·4-s − 0.883·5-s − 0.148·6-s − 1.90·7-s + 0.0430·8-s − 0.988·9-s + 1.24·10-s − 0.312·11-s + 0.102·12-s − 0.621·13-s + 2.67·14-s − 0.0935·15-s − 1.02·16-s + 0.831·17-s + 1.38·18-s + 0.229·19-s − 0.856·20-s − 0.201·21-s + 0.438·22-s + 1.47·23-s + 0.00455·24-s − 0.218·25-s + 0.871·26-s − 0.210·27-s − 1.85·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 2 | \( 1 + 1.98T + 2T^{2} \) |
| 3 | \( 1 - 0.183T + 3T^{2} \) |
| 5 | \( 1 + 1.97T + 5T^{2} \) |
| 7 | \( 1 + 5.05T + 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 + 2.23T + 13T^{2} \) |
| 17 | \( 1 - 3.42T + 17T^{2} \) |
| 23 | \( 1 - 7.05T + 23T^{2} \) |
| 29 | \( 1 + 5.16T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 1.06T + 37T^{2} \) |
| 41 | \( 1 - 2.33T + 41T^{2} \) |
| 43 | \( 1 + 7.22T + 43T^{2} \) |
| 47 | \( 1 - 6.90T + 47T^{2} \) |
| 53 | \( 1 - 3.02T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 5.86T + 61T^{2} \) |
| 67 | \( 1 - 9.91T + 67T^{2} \) |
| 71 | \( 1 - 0.176T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 4.79T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 9.93T + 97T^{2} \) |
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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
−8.239435917831175630420921384522, −7.41879699289813503303997410328, −7.00676251635043209871308075487, −6.11643591245797236961325667122, −5.22984522317719360542291948664, −4.02201354013374101056481642602, −3.10715907643923787397856117627, −2.58003074236422149458949988182, −0.818730340481411765042084388490, 0,
0.818730340481411765042084388490, 2.58003074236422149458949988182, 3.10715907643923787397856117627, 4.02201354013374101056481642602, 5.22984522317719360542291948664, 6.11643591245797236961325667122, 7.00676251635043209871308075487, 7.41879699289813503303997410328, 8.239435917831175630420921384522
