L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 12.4·5-s + 6·6-s + 33.5·7-s + 8·8-s + 9·9-s + 24.9·10-s + 44.6·11-s + 12·12-s − 59.8·13-s + 67.0·14-s + 37.3·15-s + 16·16-s − 72.8·17-s + 18·18-s + 49.8·20-s + 100.·21-s + 89.2·22-s + 157.·23-s + 24·24-s + 30.1·25-s − 119.·26-s + 27·27-s + 134.·28-s + 16.2·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.11·5-s + 0.408·6-s + 1.81·7-s + 0.353·8-s + 0.333·9-s + 0.787·10-s + 1.22·11-s + 0.288·12-s − 1.27·13-s + 1.28·14-s + 0.643·15-s + 0.250·16-s − 1.03·17-s + 0.235·18-s + 0.557·20-s + 1.04·21-s + 0.865·22-s + 1.42·23-s + 0.204·24-s + 0.241·25-s − 0.903·26-s + 0.192·27-s + 0.905·28-s + 0.104·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(7.745399159\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.745399159\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - 12.4T + 125T^{2} \) |
| 7 | \( 1 - 33.5T + 343T^{2} \) |
| 11 | \( 1 - 44.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 59.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 72.8T + 4.91e3T^{2} \) |
| 23 | \( 1 - 157.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 16.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 265.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 182.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 147.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 474.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 331.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 293.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 256.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 631.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 170.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 350.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 915.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 334.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 731.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 815.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.20e3T + 9.12e5T^{2} \) |
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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
−8.966791834033862909378655049921, −7.76772238163001528388153224795, −7.19355600823631156004195884830, −6.31028789310953207494908783476, −5.32580330435904081139639390645, −4.73021206957647948418094914009, −4.02311833693366235230496002353, −2.62645801795956667765121297013, −1.98128740973806870574953045505, −1.25025993648520812681789038341,
1.25025993648520812681789038341, 1.98128740973806870574953045505, 2.62645801795956667765121297013, 4.02311833693366235230496002353, 4.73021206957647948418094914009, 5.32580330435904081139639390645, 6.31028789310953207494908783476, 7.19355600823631156004195884830, 7.76772238163001528388153224795, 8.966791834033862909378655049921