| L(s) = 1 | + 0.820·3-s + 5-s − 7-s − 2.32·9-s + 1.23·11-s + 13-s + 0.820·15-s − 0.313·17-s − 1.90·19-s − 0.820·21-s − 7.89·23-s + 25-s − 4.36·27-s + 3.42·29-s − 6.27·31-s + 1.01·33-s − 35-s − 4.05·37-s + 0.820·39-s − 7.79·41-s + 2.87·43-s − 2.32·45-s + 11.5·47-s + 49-s − 0.256·51-s − 4.65·53-s + 1.23·55-s + ⋯ |
| L(s) = 1 | + 0.473·3-s + 0.447·5-s − 0.377·7-s − 0.775·9-s + 0.372·11-s + 0.277·13-s + 0.211·15-s − 0.0759·17-s − 0.437·19-s − 0.178·21-s − 1.64·23-s + 0.200·25-s − 0.840·27-s + 0.636·29-s − 1.12·31-s + 0.176·33-s − 0.169·35-s − 0.666·37-s + 0.131·39-s − 1.21·41-s + 0.438·43-s − 0.346·45-s + 1.69·47-s + 0.142·49-s − 0.0359·51-s − 0.639·53-s + 0.166·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.820T + 3T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 17 | \( 1 + 0.313T + 17T^{2} \) |
| 19 | \( 1 + 1.90T + 19T^{2} \) |
| 23 | \( 1 + 7.89T + 23T^{2} \) |
| 29 | \( 1 - 3.42T + 29T^{2} \) |
| 31 | \( 1 + 6.27T + 31T^{2} \) |
| 37 | \( 1 + 4.05T + 37T^{2} \) |
| 41 | \( 1 + 7.79T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 4.65T + 53T^{2} \) |
| 59 | \( 1 + 9.50T + 59T^{2} \) |
| 61 | \( 1 + 3.01T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 3.77T + 71T^{2} \) |
| 73 | \( 1 + 5.66T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.219891010179709465988701397588, −7.56986264674035254303838703362, −6.52285030471651503091136560001, −6.03640927285934350426535462415, −5.26352441550839454361465841895, −4.14667961753109269745610518733, −3.42656927291329189050385780054, −2.51020122380301799864076599162, −1.64812963396263772664863426115, 0,
1.64812963396263772664863426115, 2.51020122380301799864076599162, 3.42656927291329189050385780054, 4.14667961753109269745610518733, 5.26352441550839454361465841895, 6.03640927285934350426535462415, 6.52285030471651503091136560001, 7.56986264674035254303838703362, 8.219891010179709465988701397588
