| L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 8-s + 9-s − 3·10-s − 4·11-s + 12-s − 3·13-s + 3·15-s + 16-s − 18-s + 3·20-s + 4·22-s − 24-s + 4·25-s + 3·26-s + 27-s + 29-s − 3·30-s + 2·31-s − 32-s − 4·33-s + 36-s + 5·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.948·10-s − 1.20·11-s + 0.288·12-s − 0.832·13-s + 0.774·15-s + 1/4·16-s − 0.235·18-s + 0.670·20-s + 0.852·22-s − 0.204·24-s + 4/5·25-s + 0.588·26-s + 0.192·27-s + 0.185·29-s − 0.547·30-s + 0.359·31-s − 0.176·32-s − 0.696·33-s + 1/6·36-s + 0.821·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155526 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155526 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.399773761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.399773761\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−13.35182486694002, −12.84353655497574, −12.47314354084862, −11.95397212192834, −11.14160813780260, −10.79565276197429, −10.24846830480880, −9.867439969359299, −9.530303755145664, −9.127625291802799, −8.481044999690339, −8.002952758863265, −7.593682641886030, −7.096768516677205, −6.373094541453870, −6.102686770150868, −5.298061716791620, −5.041009000315349, −4.341673200350814, −3.459211019906476, −2.818691361682520, −2.380972291946100, −2.036538333191923, −1.276058097882385, −0.4808678143079287,
0.4808678143079287, 1.276058097882385, 2.036538333191923, 2.380972291946100, 2.818691361682520, 3.459211019906476, 4.341673200350814, 5.041009000315349, 5.298061716791620, 6.102686770150868, 6.373094541453870, 7.096768516677205, 7.593682641886030, 8.002952758863265, 8.481044999690339, 9.127625291802799, 9.530303755145664, 9.867439969359299, 10.24846830480880, 10.79565276197429, 11.14160813780260, 11.95397212192834, 12.47314354084862, 12.84353655497574, 13.35182486694002
