| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 3·11-s + 12-s + 2·13-s + 16-s − 3·17-s − 18-s + 2·19-s + 3·22-s − 23-s − 24-s − 5·25-s − 2·26-s + 27-s + 3·29-s + 2·31-s − 32-s − 3·33-s + 3·34-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 − 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s + 0.554·13-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.458·19-s + 0.639·22-s − 0.208·23-s − 0.204·24-s − 25-s − 0.392·26-s + 0.192·27-s + 0.557·29-s + 0.359·31-s − 0.176·32-s − 0.522·33-s + 0.514·34-s + 1/6·36-s + 0.328·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.564923196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.564923196\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.955472179205921746543519760773, −7.60858517872717307083864636471, −6.66691823920949203496506842104, −6.06166660779935536753584862109, −5.16923757274883599702970468320, −4.28369467296016055227937105201, −3.40725966796330949895387914006, −2.58256061943529050233320831924, −1.87387384084085622846369855037, −0.68293986877254494221991649190,
0.68293986877254494221991649190, 1.87387384084085622846369855037, 2.58256061943529050233320831924, 3.40725966796330949895387914006, 4.28369467296016055227937105201, 5.16923757274883599702970468320, 6.06166660779935536753584862109, 6.66691823920949203496506842104, 7.60858517872717307083864636471, 7.955472179205921746543519760773
