| L(s) = 1 | + 1.70·3-s − 0.637·5-s + 1.14·7-s − 0.0945·9-s + 0.485·11-s − 4.68·13-s − 1.08·15-s − 17-s + 1.90·19-s + 1.95·21-s − 0.312·23-s − 4.59·25-s − 5.27·27-s − 3.80·29-s − 1.49·31-s + 0.827·33-s − 0.731·35-s + 3.52·37-s − 7.97·39-s − 5.10·41-s − 2.74·43-s + 0.0603·45-s + 5.63·47-s − 5.68·49-s − 1.70·51-s − 5.67·53-s − 0.309·55-s + ⋯ |
| L(s) = 1 | + 0.984·3-s − 0.285·5-s + 0.433·7-s − 0.0315·9-s + 0.146·11-s − 1.29·13-s − 0.280·15-s − 0.242·17-s + 0.436·19-s + 0.426·21-s − 0.0651·23-s − 0.918·25-s − 1.01·27-s − 0.706·29-s − 0.269·31-s + 0.144·33-s − 0.123·35-s + 0.578·37-s − 1.27·39-s − 0.797·41-s − 0.417·43-s + 0.00899·45-s + 0.821·47-s − 0.811·49-s − 0.238·51-s − 0.779·53-s − 0.0417·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 1.70T + 3T^{2} \) |
| 5 | \( 1 + 0.637T + 5T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 - 0.485T + 11T^{2} \) |
| 13 | \( 1 + 4.68T + 13T^{2} \) |
| 19 | \( 1 - 1.90T + 19T^{2} \) |
| 23 | \( 1 + 0.312T + 23T^{2} \) |
| 29 | \( 1 + 3.80T + 29T^{2} \) |
| 31 | \( 1 + 1.49T + 31T^{2} \) |
| 37 | \( 1 - 3.52T + 37T^{2} \) |
| 41 | \( 1 + 5.10T + 41T^{2} \) |
| 43 | \( 1 + 2.74T + 43T^{2} \) |
| 47 | \( 1 - 5.63T + 47T^{2} \) |
| 53 | \( 1 + 5.67T + 53T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 7.64T + 67T^{2} \) |
| 71 | \( 1 - 1.03T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 0.162T + 79T^{2} \) |
| 83 | \( 1 + 1.63T + 83T^{2} \) |
| 89 | \( 1 + 18.8T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.993381611676128007587357362692, −7.61390430245834783597757060484, −6.83711501044762969249677074791, −5.77250519680818572952715210519, −5.00420920440566689328517239267, −4.15133523319918230590832948111, −3.33839979234577695578013708132, −2.50266244068807000163940849076, −1.70608737982944304768203963650, 0,
1.70608737982944304768203963650, 2.50266244068807000163940849076, 3.33839979234577695578013708132, 4.15133523319918230590832948111, 5.00420920440566689328517239267, 5.77250519680818572952715210519, 6.83711501044762969249677074791, 7.61390430245834783597757060484, 7.993381611676128007587357362692
