L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 2.44·7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 4·13-s + 2.44·14-s + 15-s + 16-s + 17-s − 18-s + 20-s − 2.44·21-s + 22-s + 2.83·23-s − 24-s + 25-s − 4·26-s + 27-s − 2.44·28-s + 2.44·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.922·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.652·14-s + 0.258·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.223·20-s − 0.532·21-s + 0.213·22-s + 0.591·23-s − 0.204·24-s + 0.200·25-s − 0.784·26-s + 0.192·27-s − 0.461·28-s + 0.453·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.817231411\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.817231411\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + 2.44T + 7T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2.83T + 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 + 5.27T + 37T^{2} \) |
| 41 | \( 1 - 7.27T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 7.27T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 5.27T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 0.396T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 1.60T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 1.11T + 89T^{2} \) |
| 97 | \( 1 - 1.23T + 97T^{2} \) |
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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
−8.196172088429186334333893552647, −7.61795148340788982532177760506, −6.63827598484555361800107812296, −6.30654037836641860395022912589, −5.42976967685011095280415251760, −4.36342356807936378204085893242, −3.31978007771212236157926726715, −2.86309119054845837324922735592, −1.79917961283997852172047623004, −0.791320477470567192007511747450,
0.791320477470567192007511747450, 1.79917961283997852172047623004, 2.86309119054845837324922735592, 3.31978007771212236157926726715, 4.36342356807936378204085893242, 5.42976967685011095280415251760, 6.30654037836641860395022912589, 6.63827598484555361800107812296, 7.61795148340788982532177760506, 8.196172088429186334333893552647