| L(s) = 1 | + 3-s − 5-s − 2.82·7-s + 9-s − 2.82·11-s + 13-s − 15-s − 3.65·17-s + 1.17·19-s − 2.82·21-s − 4·23-s + 25-s + 27-s − 7.65·29-s + 6.82·31-s − 2.82·33-s + 2.82·35-s − 2·37-s + 39-s − 2·41-s + 9.65·43-s − 45-s + 6.82·47-s + 1.00·49-s − 3.65·51-s − 2·53-s + 2.82·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.06·7-s + 0.333·9-s − 0.852·11-s + 0.277·13-s − 0.258·15-s − 0.886·17-s + 0.268·19-s − 0.617·21-s − 0.834·23-s + 0.200·25-s + 0.192·27-s − 1.42·29-s + 1.22·31-s − 0.492·33-s + 0.478·35-s − 0.328·37-s + 0.160·39-s − 0.312·41-s + 1.47·43-s − 0.149·45-s + 0.996·47-s + 0.142·49-s − 0.512·51-s − 0.274·53-s + 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.376561592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376561592\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 - 6.82T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 2.82T + 67T^{2} \) |
| 71 | \( 1 + 5.17T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 - 1.17T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−7.986670566002030683395853942275, −7.45470429801986328900002231543, −6.69353084831137070819910273190, −6.02336281164342769701599003632, −5.17551722681849086834980266010, −4.18758962183782718635967707456, −3.63002299131087803642245995445, −2.78880310297802647189884345659, −2.08485894192586346884349811534, −0.56492743521116634883363645384,
0.56492743521116634883363645384, 2.08485894192586346884349811534, 2.78880310297802647189884345659, 3.63002299131087803642245995445, 4.18758962183782718635967707456, 5.17551722681849086834980266010, 6.02336281164342769701599003632, 6.69353084831137070819910273190, 7.45470429801986328900002231543, 7.986670566002030683395853942275
