L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 6·6-s + 32·7-s − 8·8-s + 9·9-s + 50·11-s + 12·12-s + 13·13-s − 64·14-s + 16·16-s + 30·17-s − 18·18-s − 120·19-s + 96·21-s − 100·22-s + 20·23-s − 24·24-s − 26·26-s + 27·27-s + 128·28-s + 82·29-s − 44·31-s − 32·32-s + 150·33-s − 60·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.72·7-s − 0.353·8-s + 1/3·9-s + 1.37·11-s + 0.288·12-s + 0.277·13-s − 1.22·14-s + 1/4·16-s + 0.428·17-s − 0.235·18-s − 1.44·19-s + 0.997·21-s − 0.969·22-s + 0.181·23-s − 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.863·28-s + 0.525·29-s − 0.254·31-s − 0.176·32-s + 0.791·33-s − 0.302·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.083877777\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.083877777\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - p T \) |
good | 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 50 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 + 120 T + p^{3} T^{2} \) |
| 23 | \( 1 - 20 T + p^{3} T^{2} \) |
| 29 | \( 1 - 82 T + p^{3} T^{2} \) |
| 31 | \( 1 + 44 T + p^{3} T^{2} \) |
| 37 | \( 1 - 306 T + p^{3} T^{2} \) |
| 41 | \( 1 - 108 T + p^{3} T^{2} \) |
| 43 | \( 1 - 356 T + p^{3} T^{2} \) |
| 47 | \( 1 - 178 T + p^{3} T^{2} \) |
| 53 | \( 1 + 198 T + p^{3} T^{2} \) |
| 59 | \( 1 - 94 T + p^{3} T^{2} \) |
| 61 | \( 1 + 62 T + p^{3} T^{2} \) |
| 67 | \( 1 - 140 T + p^{3} T^{2} \) |
| 71 | \( 1 + 778 T + p^{3} T^{2} \) |
| 73 | \( 1 + 62 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1096 T + p^{3} T^{2} \) |
| 83 | \( 1 - 462 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1224 T + p^{3} T^{2} \) |
| 97 | \( 1 + 614 T + p^{3} T^{2} \) |
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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
−8.837564251358462232738296150643, −8.122398426060393378265499879483, −7.57667185963885325916095576406, −6.63525493653951158219266243778, −5.78512102866492684468308312124, −4.53009478092778775606530527390, −3.98237644714108410727841558131, −2.58797246449074444562702102631, −1.67225593803611380890610707635, −0.973156804073144650892901608809,
0.973156804073144650892901608809, 1.67225593803611380890610707635, 2.58797246449074444562702102631, 3.98237644714108410727841558131, 4.53009478092778775606530527390, 5.78512102866492684468308312124, 6.63525493653951158219266243778, 7.57667185963885325916095576406, 8.122398426060393378265499879483, 8.837564251358462232738296150643