| L(s) = 1 | + 0.320·2-s − 1.89·4-s − 5-s + 4.94·7-s − 1.25·8-s − 0.320·10-s − 5.87·11-s − 1.94·13-s + 1.58·14-s + 3.39·16-s − 2.22·17-s + 2.30·19-s + 1.89·20-s − 1.88·22-s − 6.59·23-s + 25-s − 0.623·26-s − 9.38·28-s + 1.58·29-s + 4.58·31-s + 3.58·32-s − 0.714·34-s − 4.94·35-s + 7.15·37-s + 0.738·38-s + 1.25·40-s + 0.124·41-s + ⋯ |
| L(s) = 1 | + 0.226·2-s − 0.948·4-s − 0.447·5-s + 1.87·7-s − 0.441·8-s − 0.101·10-s − 1.77·11-s − 0.539·13-s + 0.424·14-s + 0.848·16-s − 0.540·17-s + 0.528·19-s + 0.424·20-s − 0.402·22-s − 1.37·23-s + 0.200·25-s − 0.122·26-s − 1.77·28-s + 0.294·29-s + 0.823·31-s + 0.634·32-s − 0.122·34-s − 0.836·35-s + 1.17·37-s + 0.119·38-s + 0.197·40-s + 0.0194·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.378090556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.378090556\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 - 0.320T + 2T^{2} \) |
| 7 | \( 1 - 4.94T + 7T^{2} \) |
| 11 | \( 1 + 5.87T + 11T^{2} \) |
| 13 | \( 1 + 1.94T + 13T^{2} \) |
| 17 | \( 1 + 2.22T + 17T^{2} \) |
| 19 | \( 1 - 2.30T + 19T^{2} \) |
| 23 | \( 1 + 6.59T + 23T^{2} \) |
| 29 | \( 1 - 1.58T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 - 7.15T + 37T^{2} \) |
| 41 | \( 1 - 0.124T + 41T^{2} \) |
| 43 | \( 1 - 1.60T + 43T^{2} \) |
| 47 | \( 1 - 0.375T + 47T^{2} \) |
| 53 | \( 1 - 1.94T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 1.45T + 67T^{2} \) |
| 71 | \( 1 + 3.17T + 71T^{2} \) |
| 73 | \( 1 + 7.38T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 97 | \( 1 - 0.637T + 97T^{2} \) |
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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
−8.223496961752793435477065801348, −7.916270078322364249268036808394, −7.34921472651073783303269823142, −5.89255312577398551249828265551, −5.27957579376533659918255715522, −4.60079098393851092157154202069, −4.26155301532445865846009067011, −2.93761100936835884209460545986, −2.04660850285153222623440540625, −0.64715607121714273505890212087,
0.64715607121714273505890212087, 2.04660850285153222623440540625, 2.93761100936835884209460545986, 4.26155301532445865846009067011, 4.60079098393851092157154202069, 5.27957579376533659918255715522, 5.89255312577398551249828265551, 7.34921472651073783303269823142, 7.916270078322364249268036808394, 8.223496961752793435477065801348
