L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 4·11-s + 12-s + 6·13-s − 16-s − 4·17-s + 18-s − 2·19-s + 4·22-s − 23-s + 3·24-s − 5·25-s + 6·26-s − 27-s + 2·29-s − 4·31-s + 5·32-s − 4·33-s − 4·34-s − 36-s + 2·37-s − 2·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 1.66·13-s − 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.458·19-s + 0.852·22-s − 0.208·23-s + 0.612·24-s − 25-s + 1.17·26-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.883·32-s − 0.696·33-s − 0.685·34-s − 1/6·36-s + 0.328·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.804380550\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.804380550\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.777189075877992884842439260940, −7.932064828839316884026498283002, −6.70206404150808998760951214918, −6.23021103057897426141874753360, −5.68195104737116298433568694308, −4.64983466386812914028477867317, −3.99318231398277204106985985079, −3.51073144451088399010587234214, −2.00022392271948453847295980974, −0.75739838906060291586223443436,
0.75739838906060291586223443436, 2.00022392271948453847295980974, 3.51073144451088399010587234214, 3.99318231398277204106985985079, 4.64983466386812914028477867317, 5.68195104737116298433568694308, 6.23021103057897426141874753360, 6.70206404150808998760951214918, 7.932064828839316884026498283002, 8.777189075877992884842439260940