| L(s) = 1 | − 2.68·2-s + 0.238·3-s + 5.20·4-s − 3.86·5-s − 0.639·6-s + 2.11·7-s − 8.61·8-s − 2.94·9-s + 10.3·10-s − 3.07·11-s + 1.23·12-s − 2.57·13-s − 5.67·14-s − 0.920·15-s + 12.7·16-s + 17-s + 7.90·18-s + 3.49·19-s − 20.1·20-s + 0.503·21-s + 8.25·22-s − 4.35·23-s − 2.05·24-s + 9.95·25-s + 6.92·26-s − 1.41·27-s + 11.0·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 0.137·3-s + 2.60·4-s − 1.72·5-s − 0.260·6-s + 0.798·7-s − 3.04·8-s − 0.981·9-s + 3.28·10-s − 0.927·11-s + 0.357·12-s − 0.715·13-s − 1.51·14-s − 0.237·15-s + 3.17·16-s + 0.242·17-s + 1.86·18-s + 0.802·19-s − 4.50·20-s + 0.109·21-s + 1.76·22-s − 0.907·23-s − 0.418·24-s + 1.99·25-s + 1.35·26-s − 0.272·27-s + 2.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 353 | \( 1 - T \) |
| good | 2 | \( 1 + 2.68T + 2T^{2} \) |
| 3 | \( 1 - 0.238T + 3T^{2} \) |
| 5 | \( 1 + 3.86T + 5T^{2} \) |
| 7 | \( 1 - 2.11T + 7T^{2} \) |
| 11 | \( 1 + 3.07T + 11T^{2} \) |
| 13 | \( 1 + 2.57T + 13T^{2} \) |
| 19 | \( 1 - 3.49T + 19T^{2} \) |
| 23 | \( 1 + 4.35T + 23T^{2} \) |
| 29 | \( 1 - 2.16T + 29T^{2} \) |
| 31 | \( 1 + 0.560T + 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 + 5.50T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 1.32T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 - 9.14T + 59T^{2} \) |
| 61 | \( 1 + 0.467T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 0.828T + 79T^{2} \) |
| 83 | \( 1 - 2.00T + 83T^{2} \) |
| 89 | \( 1 - 0.0494T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.83350867889899834702459824122, −7.54110539141609740192560116747, −6.85294992053664300918401391063, −5.71660928666433651339770736266, −4.94877407111245630572416360347, −3.75019279391933343715881760351, −2.89069888383423881967628640846, −2.19036333288640682238136947099, −0.835108364424580303094990778952, 0,
0.835108364424580303094990778952, 2.19036333288640682238136947099, 2.89069888383423881967628640846, 3.75019279391933343715881760351, 4.94877407111245630572416360347, 5.71660928666433651339770736266, 6.85294992053664300918401391063, 7.54110539141609740192560116747, 7.83350867889899834702459824122
