L(s) = 1 | − 2-s − 1.12·3-s + 4-s − 3.99·5-s + 1.12·6-s + 0.851·7-s − 8-s − 1.74·9-s + 3.99·10-s + 1.40·11-s − 1.12·12-s − 5.92·13-s − 0.851·14-s + 4.48·15-s + 16-s + 1.90·17-s + 1.74·18-s + 19-s − 3.99·20-s − 0.955·21-s − 1.40·22-s − 1.62·23-s + 1.12·24-s + 10.9·25-s + 5.92·26-s + 5.31·27-s + 0.851·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.647·3-s + 0.5·4-s − 1.78·5-s + 0.458·6-s + 0.321·7-s − 0.353·8-s − 0.580·9-s + 1.26·10-s + 0.424·11-s − 0.323·12-s − 1.64·13-s − 0.227·14-s + 1.15·15-s + 0.250·16-s + 0.462·17-s + 0.410·18-s + 0.229·19-s − 0.893·20-s − 0.208·21-s − 0.300·22-s − 0.339·23-s + 0.229·24-s + 2.19·25-s + 1.16·26-s + 1.02·27-s + 0.160·28-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
\(L(1)\) |
\(\approx\) |
\(0.1339548381\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1339548381\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.12T + 3T^{2} \) |
| 5 | \( 1 + 3.99T + 5T^{2} \) |
| 7 | \( 1 - 0.851T + 7T^{2} \) |
| 11 | \( 1 - 1.40T + 11T^{2} \) |
| 13 | \( 1 + 5.92T + 13T^{2} \) |
| 17 | \( 1 - 1.90T + 17T^{2} \) |
| 23 | \( 1 + 1.62T + 23T^{2} \) |
| 29 | \( 1 + 9.90T + 29T^{2} \) |
| 31 | \( 1 - 7.86T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + 7.76T + 41T^{2} \) |
| 43 | \( 1 + 8.87T + 43T^{2} \) |
| 47 | \( 1 - 1.18T + 47T^{2} \) |
| 53 | \( 1 + 5.44T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 - 2.48T + 71T^{2} \) |
| 73 | \( 1 + 0.617T + 73T^{2} \) |
| 79 | \( 1 - 1.13T + 79T^{2} \) |
| 83 | \( 1 + 8.32T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 4.65T + 97T^{2} \) |
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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
−7.911635246367017096207297686521, −7.36480561387366125021448763642, −6.63394524700150930063985768647, −5.85992521317554331950844147442, −4.85729456577561384796824514006, −4.49099058872377732270371513063, −3.37484386760055451013842106551, −2.78467887093276705014932010149, −1.44326514726546757043405075570, −0.21490480885496398684232417228,
0.21490480885496398684232417228, 1.44326514726546757043405075570, 2.78467887093276705014932010149, 3.37484386760055451013842106551, 4.49099058872377732270371513063, 4.85729456577561384796824514006, 5.85992521317554331950844147442, 6.63394524700150930063985768647, 7.36480561387366125021448763642, 7.911635246367017096207297686521