L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 2·5-s − 6·6-s − 21·7-s + 8·8-s + 9·9-s + 4·10-s − 40·11-s − 12·12-s − 17·13-s − 42·14-s − 6·15-s + 16·16-s + 36·17-s + 18·18-s + 8·20-s + 63·21-s − 80·22-s + 74·23-s − 24·24-s − 121·25-s − 34·26-s − 27·27-s − 84·28-s − 100·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.178·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.126·10-s − 1.09·11-s − 0.288·12-s − 0.362·13-s − 0.801·14-s − 0.103·15-s + 1/4·16-s + 0.513·17-s + 0.235·18-s + 0.0894·20-s + 0.654·21-s − 0.775·22-s + 0.670·23-s − 0.204·24-s − 0.967·25-s − 0.256·26-s − 0.192·27-s − 0.566·28-s − 0.640·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.691519195\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.691519195\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 + 17 T + p^{3} T^{2} \) |
| 17 | \( 1 - 36 T + p^{3} T^{2} \) |
| 23 | \( 1 - 74 T + p^{3} T^{2} \) |
| 29 | \( 1 + 100 T + p^{3} T^{2} \) |
| 31 | \( 1 + 103 T + p^{3} T^{2} \) |
| 37 | \( 1 + 187 T + p^{3} T^{2} \) |
| 41 | \( 1 - 128 T + p^{3} T^{2} \) |
| 43 | \( 1 - 121 T + p^{3} T^{2} \) |
| 47 | \( 1 - 410 T + p^{3} T^{2} \) |
| 53 | \( 1 + 230 T + p^{3} T^{2} \) |
| 59 | \( 1 - 744 T + p^{3} T^{2} \) |
| 61 | \( 1 + 277 T + p^{3} T^{2} \) |
| 67 | \( 1 - 231 T + p^{3} T^{2} \) |
| 71 | \( 1 + 578 T + p^{3} T^{2} \) |
| 73 | \( 1 - 609 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1259 T + p^{3} T^{2} \) |
| 83 | \( 1 + 696 T + p^{3} T^{2} \) |
| 89 | \( 1 - 612 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1550 T + p^{3} T^{2} \) |
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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
−8.775856108374878531449187699331, −7.53419908156586819715333775485, −7.15951993446566316224962620705, −6.07794227738289773583626943052, −5.63811354400338697591245895685, −4.85557505169767347573218618686, −3.79195687820374784759228949371, −2.99122520060694627347371736119, −2.01833480638905861877752155659, −0.51796278256806689093994961629,
0.51796278256806689093994961629, 2.01833480638905861877752155659, 2.99122520060694627347371736119, 3.79195687820374784759228949371, 4.85557505169767347573218618686, 5.63811354400338697591245895685, 6.07794227738289773583626943052, 7.15951993446566316224962620705, 7.53419908156586819715333775485, 8.775856108374878531449187699331