| L(s) = 1 | + 2.04·2-s + 2.18·4-s + 2.05·5-s − 7-s + 0.385·8-s + 4.20·10-s − 0.528·11-s − 2.96·13-s − 2.04·14-s − 3.58·16-s + 5.27·17-s − 4.01·19-s + 4.49·20-s − 1.08·22-s − 7.89·23-s − 0.772·25-s − 6.06·26-s − 2.18·28-s − 1.95·29-s − 4.78·31-s − 8.11·32-s + 10.8·34-s − 2.05·35-s − 1.17·37-s − 8.20·38-s + 0.793·40-s + 4.96·41-s + ⋯ |
| L(s) = 1 | + 1.44·2-s + 1.09·4-s + 0.919·5-s − 0.377·7-s + 0.136·8-s + 1.33·10-s − 0.159·11-s − 0.821·13-s − 0.546·14-s − 0.896·16-s + 1.28·17-s − 0.920·19-s + 1.00·20-s − 0.230·22-s − 1.64·23-s − 0.154·25-s − 1.18·26-s − 0.413·28-s − 0.362·29-s − 0.859·31-s − 1.43·32-s + 1.85·34-s − 0.347·35-s − 0.193·37-s − 1.33·38-s + 0.125·40-s + 0.775·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 2.04T + 2T^{2} \) |
| 5 | \( 1 - 2.05T + 5T^{2} \) |
| 11 | \( 1 + 0.528T + 11T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 17 | \( 1 - 5.27T + 17T^{2} \) |
| 19 | \( 1 + 4.01T + 19T^{2} \) |
| 23 | \( 1 + 7.89T + 23T^{2} \) |
| 29 | \( 1 + 1.95T + 29T^{2} \) |
| 31 | \( 1 + 4.78T + 31T^{2} \) |
| 37 | \( 1 + 1.17T + 37T^{2} \) |
| 41 | \( 1 - 4.96T + 41T^{2} \) |
| 43 | \( 1 - 3.63T + 43T^{2} \) |
| 47 | \( 1 + 2.64T + 47T^{2} \) |
| 53 | \( 1 - 2.95T + 53T^{2} \) |
| 59 | \( 1 - 1.25T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 0.487T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 + 7.25T + 79T^{2} \) |
| 83 | \( 1 - 5.81T + 83T^{2} \) |
| 89 | \( 1 - 7.38T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−7.34075869943709875961150770511, −6.40537282308484128305853436790, −5.85437567625914374934455100829, −5.55928473475338978918859430351, −4.65132009791521198108391982334, −3.99610761124417119543404584418, −3.22137982399632929622812072773, −2.41544326732868787188713254829, −1.76246344263137157951694125715, 0,
1.76246344263137157951694125715, 2.41544326732868787188713254829, 3.22137982399632929622812072773, 3.99610761124417119543404584418, 4.65132009791521198108391982334, 5.55928473475338978918859430351, 5.85437567625914374934455100829, 6.40537282308484128305853436790, 7.34075869943709875961150770511
