| L(s) = 1 | + 3-s + 1.69·5-s + 3.04·7-s + 9-s + 5.13·11-s + 4.38·13-s + 1.69·15-s − 7.10·17-s + 7.10·19-s + 3.04·21-s − 23-s − 2.13·25-s + 27-s + 29-s + 5.13·31-s + 5.13·33-s + 5.14·35-s − 0.265·37-s + 4.38·39-s − 7.58·41-s − 3.69·43-s + 1.69·45-s − 1.55·47-s + 2.26·49-s − 7.10·51-s + 6.33·53-s + 8.69·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.756·5-s + 1.15·7-s + 0.333·9-s + 1.54·11-s + 1.21·13-s + 0.436·15-s − 1.72·17-s + 1.63·19-s + 0.664·21-s − 0.208·23-s − 0.427·25-s + 0.192·27-s + 0.185·29-s + 0.921·31-s + 0.894·33-s + 0.870·35-s − 0.0436·37-s + 0.702·39-s − 1.18·41-s − 0.563·43-s + 0.252·45-s − 0.226·47-s + 0.323·49-s − 0.995·51-s + 0.870·53-s + 1.17·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.398839317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.398839317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 1.69T + 5T^{2} \) |
| 7 | \( 1 - 3.04T + 7T^{2} \) |
| 11 | \( 1 - 5.13T + 11T^{2} \) |
| 13 | \( 1 - 4.38T + 13T^{2} \) |
| 17 | \( 1 + 7.10T + 17T^{2} \) |
| 19 | \( 1 - 7.10T + 19T^{2} \) |
| 31 | \( 1 - 5.13T + 31T^{2} \) |
| 37 | \( 1 + 0.265T + 37T^{2} \) |
| 41 | \( 1 + 7.58T + 41T^{2} \) |
| 43 | \( 1 + 3.69T + 43T^{2} \) |
| 47 | \( 1 + 1.55T + 47T^{2} \) |
| 53 | \( 1 - 6.33T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 7.54T + 61T^{2} \) |
| 67 | \( 1 + 0.875T + 67T^{2} \) |
| 71 | \( 1 - 1.43T + 71T^{2} \) |
| 73 | \( 1 + 5.91T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 9.39T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 3.58T + 97T^{2} \) |
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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
−8.107651022973231772691281958457, −6.94329254862386620952409589924, −6.63602425937623208844413291073, −5.75725403473498924953863455843, −5.01415091502183612594349688066, −4.17041480035377403303519381328, −3.64367154678039315741539571277, −2.52845404969542994244288105961, −1.65435144872570847540421552879, −1.18069359309270916789669577897,
1.18069359309270916789669577897, 1.65435144872570847540421552879, 2.52845404969542994244288105961, 3.64367154678039315741539571277, 4.17041480035377403303519381328, 5.01415091502183612594349688066, 5.75725403473498924953863455843, 6.63602425937623208844413291073, 6.94329254862386620952409589924, 8.107651022973231772691281958457
