L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s − 4·11-s + 12-s + 2·13-s + 15-s + 16-s + 6·17-s − 18-s + 20-s + 4·22-s − 8·23-s − 24-s + 25-s − 2·26-s + 27-s + 10·29-s − 30-s + 8·31-s − 32-s − 4·33-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.223·20-s + 0.852·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 1.85·29-s − 0.182·30-s + 1.43·31-s − 0.176·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.641843825\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.641843825\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.672947411410494461714701963855, −8.474423883679326843484023496736, −8.144159451058535768339741415787, −7.37364686961163198264124371321, −6.26480283429435197593792899951, −5.58313671388909558006581635222, −4.37063079475878442176743330274, −3.10205830875256288401246717147, −2.35089399689505960435017244261, −1.03116026694611315150632321310,
1.03116026694611315150632321310, 2.35089399689505960435017244261, 3.10205830875256288401246717147, 4.37063079475878442176743330274, 5.58313671388909558006581635222, 6.26480283429435197593792899951, 7.37364686961163198264124371321, 8.144159451058535768339741415787, 8.474423883679326843484023496736, 9.672947411410494461714701963855