| L(s) = 1 | + 1.41·3-s − 1.10·5-s − 3.91·7-s − 1.00·9-s − 11-s + 2.55·13-s − 1.55·15-s − 2.06·17-s + 4.47·19-s − 5.52·21-s + 2.63·23-s − 3.77·25-s − 5.65·27-s − 7.02·29-s − 2.21·31-s − 1.41·33-s + 4.32·35-s + 7.48·37-s + 3.61·39-s + 10.6·41-s − 5.52·43-s + 1.11·45-s + 3.99·47-s + 8.30·49-s − 2.91·51-s − 10.4·53-s + 1.10·55-s + ⋯ |
| L(s) = 1 | + 0.814·3-s − 0.494·5-s − 1.47·7-s − 0.336·9-s − 0.301·11-s + 0.709·13-s − 0.402·15-s − 0.500·17-s + 1.02·19-s − 1.20·21-s + 0.550·23-s − 0.755·25-s − 1.08·27-s − 1.30·29-s − 0.397·31-s − 0.245·33-s + 0.731·35-s + 1.23·37-s + 0.578·39-s + 1.67·41-s − 0.842·43-s + 0.166·45-s + 0.582·47-s + 1.18·49-s − 0.407·51-s − 1.43·53-s + 0.149·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.436546492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.436546492\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + 1.10T + 5T^{2} \) |
| 7 | \( 1 + 3.91T + 7T^{2} \) |
| 13 | \( 1 - 2.55T + 13T^{2} \) |
| 17 | \( 1 + 2.06T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 - 2.63T + 23T^{2} \) |
| 29 | \( 1 + 7.02T + 29T^{2} \) |
| 31 | \( 1 + 2.21T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 5.52T + 43T^{2} \) |
| 47 | \( 1 - 3.99T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 5.63T + 59T^{2} \) |
| 61 | \( 1 - 1.62T + 61T^{2} \) |
| 67 | \( 1 - 2.07T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 8.54T + 73T^{2} \) |
| 79 | \( 1 + 1.97T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 1.25T + 97T^{2} \) |
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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
−7.87192372271117218183355930186, −7.66660046359855117163359533899, −6.65510189570379405385768611040, −6.03222835204210951717591764289, −5.31429575885090117749615897576, −4.09988834531957825306508296069, −3.48731287666150781352356018588, −2.99404026444992338084265440144, −2.08955260984609063691315667972, −0.57487762398438821670356513732,
0.57487762398438821670356513732, 2.08955260984609063691315667972, 2.99404026444992338084265440144, 3.48731287666150781352356018588, 4.09988834531957825306508296069, 5.31429575885090117749615897576, 6.03222835204210951717591764289, 6.65510189570379405385768611040, 7.66660046359855117163359533899, 7.87192372271117218183355930186
