L(s) = 1 | + 2-s + 4-s − 2.19·5-s − 7-s + 8-s − 2.19·10-s + 1.19·11-s − 6.38·13-s − 14-s + 16-s + 4·17-s + 19-s − 2.19·20-s + 1.19·22-s + 8.38·23-s − 0.192·25-s − 6.38·26-s − 28-s + 0.192·29-s + 10·31-s + 32-s + 4·34-s + 2.19·35-s + 4.19·37-s + 38-s − 2.19·40-s − 6.57·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.980·5-s − 0.377·7-s + 0.353·8-s − 0.693·10-s + 0.359·11-s − 1.77·13-s − 0.267·14-s + 0.250·16-s + 0.970·17-s + 0.229·19-s − 0.490·20-s + 0.254·22-s + 1.74·23-s − 0.0385·25-s − 1.25·26-s − 0.188·28-s + 0.0357·29-s + 1.79·31-s + 0.176·32-s + 0.685·34-s + 0.370·35-s + 0.689·37-s + 0.162·38-s − 0.346·40-s − 1.02·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.173457269\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.173457269\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 2.19T + 5T^{2} \) |
| 11 | \( 1 - 1.19T + 11T^{2} \) |
| 13 | \( 1 + 6.38T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 23 | \( 1 - 8.38T + 23T^{2} \) |
| 29 | \( 1 - 0.192T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 - 4.19T + 37T^{2} \) |
| 41 | \( 1 + 6.57T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 4.80T + 47T^{2} \) |
| 53 | \( 1 - 5.19T + 53T^{2} \) |
| 59 | \( 1 - 4.57T + 59T^{2} \) |
| 61 | \( 1 + 0.807T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 1.19T + 71T^{2} \) |
| 73 | \( 1 + 4.38T + 73T^{2} \) |
| 79 | \( 1 - 1.80T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 2.42T + 89T^{2} \) |
| 97 | \( 1 - 8.80T + 97T^{2} \) |
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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
−8.966008242149059243322561278381, −7.920761159820696974738027029334, −7.34983570266189996374171002122, −6.77500045012533759583742614552, −5.70231906285146580123494284112, −4.86849504731556748424436028645, −4.21370243555170539445478711553, −3.21991039346502037772994414365, −2.55405266092204699734770178641, −0.850822208850626494616504195114,
0.850822208850626494616504195114, 2.55405266092204699734770178641, 3.21991039346502037772994414365, 4.21370243555170539445478711553, 4.86849504731556748424436028645, 5.70231906285146580123494284112, 6.77500045012533759583742614552, 7.34983570266189996374171002122, 7.920761159820696974738027029334, 8.966008242149059243322561278381