| L(s) = 1 | − 0.907·2-s − 1.17·4-s − 3.19·5-s + 3.56·7-s + 2.88·8-s + 2.89·10-s + 3.27·11-s + 5.58·13-s − 3.23·14-s − 0.259·16-s − 4.23·19-s + 3.75·20-s − 2.97·22-s + 4.60·23-s + 5.19·25-s − 5.06·26-s − 4.19·28-s − 2.08·29-s + 0.448·31-s − 5.52·32-s − 11.3·35-s − 0.742·37-s + 3.84·38-s − 9.20·40-s + 4.49·41-s + 6.10·43-s − 3.85·44-s + ⋯ |
| L(s) = 1 | − 0.641·2-s − 0.588·4-s − 1.42·5-s + 1.34·7-s + 1.01·8-s + 0.915·10-s + 0.987·11-s + 1.54·13-s − 0.863·14-s − 0.0649·16-s − 0.971·19-s + 0.840·20-s − 0.633·22-s + 0.959·23-s + 1.03·25-s − 0.993·26-s − 0.792·28-s − 0.387·29-s + 0.0806·31-s − 0.977·32-s − 1.92·35-s − 0.122·37-s + 0.623·38-s − 1.45·40-s + 0.701·41-s + 0.931·43-s − 0.581·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.085909696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.085909696\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.907T + 2T^{2} \) |
| 5 | \( 1 + 3.19T + 5T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 - 3.27T + 11T^{2} \) |
| 13 | \( 1 - 5.58T + 13T^{2} \) |
| 19 | \( 1 + 4.23T + 19T^{2} \) |
| 23 | \( 1 - 4.60T + 23T^{2} \) |
| 29 | \( 1 + 2.08T + 29T^{2} \) |
| 31 | \( 1 - 0.448T + 31T^{2} \) |
| 37 | \( 1 + 0.742T + 37T^{2} \) |
| 41 | \( 1 - 4.49T + 41T^{2} \) |
| 43 | \( 1 - 6.10T + 43T^{2} \) |
| 47 | \( 1 - 2.26T + 47T^{2} \) |
| 53 | \( 1 + 7.55T + 53T^{2} \) |
| 59 | \( 1 - 2.83T + 59T^{2} \) |
| 61 | \( 1 + 3.91T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 3.64T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 5.63T + 79T^{2} \) |
| 83 | \( 1 - 3.92T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 0.828T + 97T^{2} \) |
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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
−8.777765156838135883776844300643, −8.195770983933897074028201567085, −7.70179794127564305091443582213, −6.86282875970105653525873214476, −5.76199135286255374757819020996, −4.56378336089798763783267612424, −4.24211224873688422271415065012, −3.44487857661070500240694711780, −1.65260959176254934205208701466, −0.798354625028308933122316813468,
0.798354625028308933122316813468, 1.65260959176254934205208701466, 3.44487857661070500240694711780, 4.24211224873688422271415065012, 4.56378336089798763783267612424, 5.76199135286255374757819020996, 6.86282875970105653525873214476, 7.70179794127564305091443582213, 8.195770983933897074028201567085, 8.777765156838135883776844300643
