| L(s) = 1 | + 1.34·2-s + 1.23·3-s − 0.203·4-s + 5-s + 1.66·6-s − 3.86·7-s − 2.95·8-s − 1.46·9-s + 1.34·10-s + 5.03·11-s − 0.252·12-s + 3.21·13-s − 5.17·14-s + 1.23·15-s − 3.55·16-s − 5.75·17-s − 1.96·18-s + 4.82·19-s − 0.203·20-s − 4.78·21-s + 6.74·22-s − 3.49·23-s − 3.65·24-s + 25-s + 4.30·26-s − 5.53·27-s + 0.787·28-s + ⋯ |
| L(s) = 1 | + 0.947·2-s + 0.715·3-s − 0.101·4-s + 0.447·5-s + 0.677·6-s − 1.46·7-s − 1.04·8-s − 0.488·9-s + 0.423·10-s + 1.51·11-s − 0.0729·12-s + 0.891·13-s − 1.38·14-s + 0.319·15-s − 0.887·16-s − 1.39·17-s − 0.462·18-s + 1.10·19-s − 0.0455·20-s − 1.04·21-s + 1.43·22-s − 0.728·23-s − 0.747·24-s + 0.200·25-s + 0.845·26-s − 1.06·27-s + 0.148·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 1.34T + 2T^{2} \) |
| 3 | \( 1 - 1.23T + 3T^{2} \) |
| 7 | \( 1 + 3.86T + 7T^{2} \) |
| 11 | \( 1 - 5.03T + 11T^{2} \) |
| 13 | \( 1 - 3.21T + 13T^{2} \) |
| 17 | \( 1 + 5.75T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 + 3.49T + 23T^{2} \) |
| 31 | \( 1 + 3.10T + 31T^{2} \) |
| 37 | \( 1 - 2.71T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 + 8.64T + 43T^{2} \) |
| 47 | \( 1 - 0.222T + 47T^{2} \) |
| 53 | \( 1 + 6.68T + 53T^{2} \) |
| 59 | \( 1 + 4.63T + 59T^{2} \) |
| 61 | \( 1 + 9.46T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 7.50T + 71T^{2} \) |
| 73 | \( 1 - 3.53T + 73T^{2} \) |
| 79 | \( 1 + 4.45T + 79T^{2} \) |
| 83 | \( 1 - 8.43T + 83T^{2} \) |
| 89 | \( 1 - 9.68T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−8.275171829199918371665740362497, −6.95993713598510300003836624216, −6.30294976828677517595916769345, −6.03040635807666292695384059373, −4.98034313660634558272452439237, −3.95693038061179539488680551973, −3.45515359303753619377049891609, −2.91698449877499855988588740568, −1.69925728561776504397360537243, 0,
1.69925728561776504397360537243, 2.91698449877499855988588740568, 3.45515359303753619377049891609, 3.95693038061179539488680551973, 4.98034313660634558272452439237, 6.03040635807666292695384059373, 6.30294976828677517595916769345, 6.95993713598510300003836624216, 8.275171829199918371665740362497
