| L(s) = 1 | + 2-s + 2.52·3-s + 4-s + 3.27·5-s + 2.52·6-s + 8-s + 3.38·9-s + 3.27·10-s + 1.94·11-s + 2.52·12-s − 5.01·13-s + 8.26·15-s + 16-s − 17-s + 3.38·18-s + 3.18·19-s + 3.27·20-s + 1.94·22-s − 4.62·23-s + 2.52·24-s + 5.70·25-s − 5.01·26-s + 0.969·27-s − 9.02·29-s + 8.26·30-s − 9.48·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.45·3-s + 0.5·4-s + 1.46·5-s + 1.03·6-s + 0.353·8-s + 1.12·9-s + 1.03·10-s + 0.585·11-s + 0.729·12-s − 1.38·13-s + 2.13·15-s + 0.250·16-s − 0.242·17-s + 0.797·18-s + 0.731·19-s + 0.731·20-s + 0.413·22-s − 0.964·23-s + 0.515·24-s + 1.14·25-s − 0.982·26-s + 0.186·27-s − 1.67·29-s + 1.50·30-s − 1.70·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.317222434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.317222434\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2.52T + 3T^{2} \) |
| 5 | \( 1 - 3.27T + 5T^{2} \) |
| 11 | \( 1 - 1.94T + 11T^{2} \) |
| 13 | \( 1 + 5.01T + 13T^{2} \) |
| 19 | \( 1 - 3.18T + 19T^{2} \) |
| 23 | \( 1 + 4.62T + 23T^{2} \) |
| 29 | \( 1 + 9.02T + 29T^{2} \) |
| 31 | \( 1 + 9.48T + 31T^{2} \) |
| 37 | \( 1 + 6.14T + 37T^{2} \) |
| 41 | \( 1 - 4.74T + 41T^{2} \) |
| 43 | \( 1 - 8.19T + 43T^{2} \) |
| 47 | \( 1 - 4.01T + 47T^{2} \) |
| 53 | \( 1 + 5.82T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 0.00665T + 61T^{2} \) |
| 67 | \( 1 + 8.71T + 67T^{2} \) |
| 71 | \( 1 + 2.99T + 71T^{2} \) |
| 73 | \( 1 - 8.54T + 73T^{2} \) |
| 79 | \( 1 + 6.37T + 79T^{2} \) |
| 83 | \( 1 - 2.90T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 19.6T + 97T^{2} \) |
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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
−9.455928291086351933651038670330, −8.786056827389105708585973948433, −7.53982516225809509043608595912, −7.16746619875842425437015736562, −5.92876104500388488969981872681, −5.35163772184591276704821129881, −4.17238940508891452111961665077, −3.29811216079064940209588188866, −2.24680955743008706005426660423, −1.86235325951465555756221519673,
1.86235325951465555756221519673, 2.24680955743008706005426660423, 3.29811216079064940209588188866, 4.17238940508891452111961665077, 5.35163772184591276704821129881, 5.92876104500388488969981872681, 7.16746619875842425437015736562, 7.53982516225809509043608595912, 8.786056827389105708585973948433, 9.455928291086351933651038670330
