| L(s) = 1 | − 2-s + 2.91·3-s + 4-s + 3.03·5-s − 2.91·6-s − 4.14·7-s − 8-s + 5.49·9-s − 3.03·10-s − 0.192·11-s + 2.91·12-s − 2.27·13-s + 4.14·14-s + 8.85·15-s + 16-s − 0.760·17-s − 5.49·18-s + 19-s + 3.03·20-s − 12.0·21-s + 0.192·22-s − 0.638·23-s − 2.91·24-s + 4.23·25-s + 2.27·26-s + 7.27·27-s − 4.14·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.68·3-s + 0.5·4-s + 1.35·5-s − 1.18·6-s − 1.56·7-s − 0.353·8-s + 1.83·9-s − 0.961·10-s − 0.0580·11-s + 0.841·12-s − 0.630·13-s + 1.10·14-s + 2.28·15-s + 0.250·16-s − 0.184·17-s − 1.29·18-s + 0.229·19-s + 0.679·20-s − 2.63·21-s + 0.0410·22-s − 0.133·23-s − 0.594·24-s + 0.847·25-s + 0.445·26-s + 1.40·27-s − 0.783·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.125552555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.125552555\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 2.91T + 3T^{2} \) |
| 5 | \( 1 - 3.03T + 5T^{2} \) |
| 7 | \( 1 + 4.14T + 7T^{2} \) |
| 11 | \( 1 + 0.192T + 11T^{2} \) |
| 13 | \( 1 + 2.27T + 13T^{2} \) |
| 17 | \( 1 + 0.760T + 17T^{2} \) |
| 23 | \( 1 + 0.638T + 23T^{2} \) |
| 29 | \( 1 - 8.98T + 29T^{2} \) |
| 31 | \( 1 + 0.558T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 1.60T + 41T^{2} \) |
| 43 | \( 1 + 4.44T + 43T^{2} \) |
| 47 | \( 1 - 8.30T + 47T^{2} \) |
| 53 | \( 1 - 6.71T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 16.1T + 67T^{2} \) |
| 71 | \( 1 + 0.760T + 71T^{2} \) |
| 73 | \( 1 + 7.66T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 4.17T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 3.66T + 97T^{2} \) |
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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
−7.86862775472638790740874560537, −7.34704989871477528422484193636, −6.43199886845103772289788085660, −6.21475849249500472797829510716, −5.05422735331682118682156683071, −3.97010240414749441657165898019, −3.08428983088792703376665261889, −2.58747517591246495686067193669, −2.08019659556184290770282312326, −0.885241802756900469973730211109,
0.885241802756900469973730211109, 2.08019659556184290770282312326, 2.58747517591246495686067193669, 3.08428983088792703376665261889, 3.97010240414749441657165898019, 5.05422735331682118682156683071, 6.21475849249500472797829510716, 6.43199886845103772289788085660, 7.34704989871477528422484193636, 7.86862775472638790740874560537
