| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 4·11-s − 12-s − 2·13-s + 14-s − 15-s + 16-s − 2·17-s − 18-s − 4·19-s + 20-s + 21-s − 4·22-s − 8·23-s + 24-s + 25-s + 2·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-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 + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-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 − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.73751893996795, −12.96002158607986, −12.65139232573992, −12.11567105329621, −11.78597178064950, −11.24807774360114, −10.78012725269401, −10.15633736093034, −9.996495131187522, −9.414216413950821, −8.856179183151821, −8.650366329021349, −7.841208789859405, −7.324645261685461, −6.876822164674806, −6.337351206569285, −5.969021955011886, −5.636625839903854, −4.644569562414045, −4.287471938961240, −3.761944189993901, −2.897585838395784, −2.357547746238557, −1.614819302621079, −1.299605744807361, 0, 0,
1.299605744807361, 1.614819302621079, 2.357547746238557, 2.897585838395784, 3.761944189993901, 4.287471938961240, 4.644569562414045, 5.636625839903854, 5.969021955011886, 6.337351206569285, 6.876822164674806, 7.324645261685461, 7.841208789859405, 8.650366329021349, 8.856179183151821, 9.414216413950821, 9.996495131187522, 10.15633736093034, 10.78012725269401, 11.24807774360114, 11.78597178064950, 12.11567105329621, 12.65139232573992, 12.96002158607986, 13.73751893996795
