| L(s) = 1 | + 2-s − 0.386·3-s + 4-s − 4.13·5-s − 0.386·6-s + 8-s − 2.85·9-s − 4.13·10-s − 2.54·11-s − 0.386·12-s − 3.74·13-s + 1.59·15-s + 16-s − 0.288·17-s − 2.85·18-s − 5.58·19-s − 4.13·20-s − 2.54·22-s − 7.65·23-s − 0.386·24-s + 12.1·25-s − 3.74·26-s + 2.25·27-s + 8.01·29-s + 1.59·30-s + 0.278·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.222·3-s + 0.5·4-s − 1.85·5-s − 0.157·6-s + 0.353·8-s − 0.950·9-s − 1.30·10-s − 0.768·11-s − 0.111·12-s − 1.03·13-s + 0.412·15-s + 0.250·16-s − 0.0699·17-s − 0.671·18-s − 1.28·19-s − 0.925·20-s − 0.543·22-s − 1.59·23-s − 0.0788·24-s + 2.42·25-s − 0.734·26-s + 0.434·27-s + 1.48·29-s + 0.291·30-s + 0.0500·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7975260784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7975260784\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 0.386T + 3T^{2} \) |
| 5 | \( 1 + 4.13T + 5T^{2} \) |
| 11 | \( 1 + 2.54T + 11T^{2} \) |
| 13 | \( 1 + 3.74T + 13T^{2} \) |
| 17 | \( 1 + 0.288T + 17T^{2} \) |
| 19 | \( 1 + 5.58T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 - 8.01T + 29T^{2} \) |
| 31 | \( 1 - 0.278T + 31T^{2} \) |
| 37 | \( 1 - 2.10T + 37T^{2} \) |
| 43 | \( 1 - 4.73T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 9.24T + 53T^{2} \) |
| 59 | \( 1 + 7.00T + 59T^{2} \) |
| 61 | \( 1 - 5.34T + 61T^{2} \) |
| 67 | \( 1 + 2.45T + 67T^{2} \) |
| 71 | \( 1 - 5.47T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 3.37T + 89T^{2} \) |
| 97 | \( 1 - 0.548T + 97T^{2} \) |
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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
−8.272905042325201010107490261168, −7.68987785255924120321141676914, −7.04803920653294935508242805914, −6.15949305388625330547935365689, −5.36543219769872638615488248564, −4.41331237017619789772960810431, −4.14490484664713982149519305908, −2.98389768710218803806268512042, −2.41494492841357227814792592968, −0.43142285455131757426260722239,
0.43142285455131757426260722239, 2.41494492841357227814792592968, 2.98389768710218803806268512042, 4.14490484664713982149519305908, 4.41331237017619789772960810431, 5.36543219769872638615488248564, 6.15949305388625330547935365689, 7.04803920653294935508242805914, 7.68987785255924120321141676914, 8.272905042325201010107490261168
