L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s − 12-s + 2·13-s + 4·14-s + 16-s + 6·17-s − 18-s + 4·19-s + 4·21-s + 24-s − 2·26-s − 27-s − 4·28-s + 6·29-s + 8·31-s − 32-s − 6·34-s + 36-s − 2·37-s − 4·38-s − 2·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.872·21-s + 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.755·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s − 0.648·38-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.249580483\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.249580483\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.03861937003388, −15.54660189111225, −14.87004654794901, −14.02957660953105, −13.59839499052915, −12.90579325952778, −12.32650630109291, −11.92812368866272, −11.40417875812034, −10.50999460476395, −10.15011437304970, −9.729412769791873, −9.202927012671533, −8.386603002779213, −7.866523929082841, −7.052771055686116, −6.651743730495271, −5.987679731324861, −5.558606971700793, −4.669498826075982, −3.627100755345589, −3.210986882381885, −2.421525126193676, −1.120631474017013, −0.6624214956392276,
0.6624214956392276, 1.120631474017013, 2.421525126193676, 3.210986882381885, 3.627100755345589, 4.669498826075982, 5.558606971700793, 5.987679731324861, 6.651743730495271, 7.052771055686116, 7.866523929082841, 8.386603002779213, 9.202927012671533, 9.729412769791873, 10.15011437304970, 10.50999460476395, 11.40417875812034, 11.92812368866272, 12.32650630109291, 12.90579325952778, 13.59839499052915, 14.02957660953105, 14.87004654794901, 15.54660189111225, 16.03861937003388