| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2.86·7-s + 8-s + 9-s − 3.52·11-s + 12-s + 2·13-s + 2.86·14-s + 16-s − 17-s + 18-s + 5.52·19-s + 2.86·21-s − 3.52·22-s − 8.11·23-s + 24-s + 2·26-s + 27-s + 2.86·28-s + 7.91·29-s + 10.3·31-s + 32-s − 3.52·33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.08·7-s + 0.353·8-s + 0.333·9-s − 1.06·11-s + 0.288·12-s + 0.554·13-s + 0.765·14-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 1.26·19-s + 0.625·21-s − 0.751·22-s − 1.69·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.541·28-s + 1.46·29-s + 1.86·31-s + 0.176·32-s − 0.613·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.060281258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.060281258\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2.86T + 7T^{2} \) |
| 11 | \( 1 + 3.52T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 - 5.52T + 19T^{2} \) |
| 23 | \( 1 + 8.11T + 23T^{2} \) |
| 29 | \( 1 - 7.91T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 6.65T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 1.52T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 0.270T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 0.929T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 4.98T + 73T^{2} \) |
| 79 | \( 1 + 2.38T + 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 + 9.25T + 89T^{2} \) |
| 97 | \( 1 - 0.747T + 97T^{2} \) |
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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
−8.599643364264755428601025495790, −8.032704790450536681769506666204, −7.59345259017797224607823365961, −6.49351205500752917083177211073, −5.67986377980204014017302071509, −4.80714078959518418591562198702, −4.24389348300421225522518486228, −3.10053413841618940500700638905, −2.36662056846552086665052705942, −1.23976829639127030474487671346,
1.23976829639127030474487671346, 2.36662056846552086665052705942, 3.10053413841618940500700638905, 4.24389348300421225522518486228, 4.80714078959518418591562198702, 5.67986377980204014017302071509, 6.49351205500752917083177211073, 7.59345259017797224607823365961, 8.032704790450536681769506666204, 8.599643364264755428601025495790
