L(s) = 1 | + 2-s − 2.85·3-s + 4-s + 2.60·5-s − 2.85·6-s − 2.89·7-s + 8-s + 5.17·9-s + 2.60·10-s + 2.99·11-s − 2.85·12-s − 4.16·13-s − 2.89·14-s − 7.43·15-s + 16-s + 2.55·17-s + 5.17·18-s + 4.52·19-s + 2.60·20-s + 8.26·21-s + 2.99·22-s + 2.80·23-s − 2.85·24-s + 1.76·25-s − 4.16·26-s − 6.21·27-s − 2.89·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.65·3-s + 0.5·4-s + 1.16·5-s − 1.16·6-s − 1.09·7-s + 0.353·8-s + 1.72·9-s + 0.822·10-s + 0.902·11-s − 0.825·12-s − 1.15·13-s − 0.772·14-s − 1.91·15-s + 0.250·16-s + 0.620·17-s + 1.21·18-s + 1.03·19-s + 0.581·20-s + 1.80·21-s + 0.638·22-s + 0.585·23-s − 0.583·24-s + 0.352·25-s − 0.816·26-s − 1.19·27-s − 0.546·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.973634191\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.973634191\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
good | 3 | \( 1 + 2.85T + 3T^{2} \) |
| 5 | \( 1 - 2.60T + 5T^{2} \) |
| 7 | \( 1 + 2.89T + 7T^{2} \) |
| 11 | \( 1 - 2.99T + 11T^{2} \) |
| 13 | \( 1 + 4.16T + 13T^{2} \) |
| 17 | \( 1 - 2.55T + 17T^{2} \) |
| 19 | \( 1 - 4.52T + 19T^{2} \) |
| 23 | \( 1 - 2.80T + 23T^{2} \) |
| 29 | \( 1 + 3.38T + 29T^{2} \) |
| 37 | \( 1 + 1.20T + 37T^{2} \) |
| 41 | \( 1 + 4.87T + 41T^{2} \) |
| 43 | \( 1 - 4.40T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 1.01T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 5.98T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 8.26T + 83T^{2} \) |
| 89 | \( 1 + 6.86T + 89T^{2} \) |
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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
−7.61530683661746085503014627772, −6.93733303009595167079648971213, −6.47262002149544497058384296617, −5.83430526212176122433900968932, −5.38350084086590341405436671832, −4.77827038656484457558781080515, −3.76885256926093133481128017127, −2.87095484129621821834404723192, −1.75878242747728436630095739431, −0.73454047081618681510853452639,
0.73454047081618681510853452639, 1.75878242747728436630095739431, 2.87095484129621821834404723192, 3.76885256926093133481128017127, 4.77827038656484457558781080515, 5.38350084086590341405436671832, 5.83430526212176122433900968932, 6.47262002149544497058384296617, 6.93733303009595167079648971213, 7.61530683661746085503014627772