L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 3·7-s − 8-s + 9-s + 10-s − 12-s − 13-s + 3·14-s + 15-s + 16-s + 7·17-s − 18-s + 5·19-s − 20-s + 3·21-s − 8·23-s + 24-s + 25-s + 26-s − 27-s − 3·28-s + 5·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.277·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s + 1.14·19-s − 0.223·20-s + 0.654·21-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.566·28-s + 0.928·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5974067793\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5974067793\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.434619325514851110490555148572, −7.77049789872902778638533918466, −7.17597688792461810600841461991, −6.36411715944922490710130608918, −5.73696462087675392990589139672, −4.89758905800619126505365147612, −3.58345532764719452575260203850, −3.17534707682927881184223136074, −1.73080658302757130293409278688, −0.51546057438666302322702554102,
0.51546057438666302322702554102, 1.73080658302757130293409278688, 3.17534707682927881184223136074, 3.58345532764719452575260203850, 4.89758905800619126505365147612, 5.73696462087675392990589139672, 6.36411715944922490710130608918, 7.17597688792461810600841461991, 7.77049789872902778638533918466, 8.434619325514851110490555148572