| L(s) = 1 | − 5.09·2-s − 5·3-s + 17.9·4-s + 5·5-s + 25.4·6-s − 20.3·7-s − 50.9·8-s − 2·9-s − 25.4·10-s − 89.9·12-s − 61.1·13-s + 103.·14-s − 25·15-s + 115.·16-s + 20.3·17-s + 10.1·18-s + 101.·19-s + 89.9·20-s + 101.·21-s + 35·23-s + 254.·24-s − 100·25-s + 312·26-s + 145·27-s − 367.·28-s + 203.·29-s + 127.·30-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 0.962·3-s + 2.24·4-s + 0.447·5-s + 1.73·6-s − 1.10·7-s − 2.25·8-s − 0.0740·9-s − 0.806·10-s − 2.16·12-s − 1.30·13-s + 1.98·14-s − 0.430·15-s + 1.81·16-s + 0.290·17-s + 0.133·18-s + 1.23·19-s + 1.00·20-s + 1.05·21-s + 0.317·23-s + 2.16·24-s − 0.800·25-s + 2.35·26-s + 1.03·27-s − 2.47·28-s + 1.30·29-s + 0.775·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3423841462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3423841462\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + 5.09T + 8T^{2} \) |
| 3 | \( 1 + 5T + 27T^{2} \) |
| 5 | \( 1 - 5T + 125T^{2} \) |
| 7 | \( 1 + 20.3T + 343T^{2} \) |
| 13 | \( 1 + 61.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 35T + 1.21e4T^{2} \) |
| 29 | \( 1 - 203.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 15T + 2.97e4T^{2} \) |
| 37 | \( 1 + 265T + 5.06e4T^{2} \) |
| 41 | \( 1 - 101.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 448.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 380T + 1.03e5T^{2} \) |
| 53 | \( 1 - 510T + 1.48e5T^{2} \) |
| 59 | \( 1 - 21T + 2.05e5T^{2} \) |
| 61 | \( 1 + 203.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 585T + 3.00e5T^{2} \) |
| 71 | \( 1 - 313T + 3.57e5T^{2} \) |
| 73 | \( 1 + 469.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 611.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 652.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 185T + 7.04e5T^{2} \) |
| 97 | \( 1 - 785T + 9.12e5T^{2} \) |
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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
−12.38444313167055396721183595283, −11.73308216932898227596465506185, −10.48905755569781590256203386280, −9.865596332916978634216672094295, −9.031128309241748518222764452641, −7.53668951669615441983053747295, −6.60929075855108328594088789092, −5.52270841248317207014064366443, −2.68444066643491194501825164513, −0.65662300966691295120940804035,
0.65662300966691295120940804035, 2.68444066643491194501825164513, 5.52270841248317207014064366443, 6.60929075855108328594088789092, 7.53668951669615441983053747295, 9.031128309241748518222764452641, 9.865596332916978634216672094295, 10.48905755569781590256203386280, 11.73308216932898227596465506185, 12.38444313167055396721183595283
