| L(s) = 1 | + 2.63·2-s − 3-s + 4.95·4-s + 5-s − 2.63·6-s + 7-s + 7.77·8-s + 9-s + 2.63·10-s + 11-s − 4.95·12-s − 3.14·13-s + 2.63·14-s − 15-s + 10.6·16-s + 0.677·17-s + 2.63·18-s + 1.95·19-s + 4.95·20-s − 21-s + 2.63·22-s + 0.293·23-s − 7.77·24-s + 25-s − 8.28·26-s − 27-s + 4.95·28-s + ⋯ |
| L(s) = 1 | + 1.86·2-s − 0.577·3-s + 2.47·4-s + 0.447·5-s − 1.07·6-s + 0.377·7-s + 2.75·8-s + 0.333·9-s + 0.833·10-s + 0.301·11-s − 1.42·12-s − 0.871·13-s + 0.704·14-s − 0.258·15-s + 2.65·16-s + 0.164·17-s + 0.621·18-s + 0.447·19-s + 1.10·20-s − 0.218·21-s + 0.562·22-s + 0.0612·23-s − 1.58·24-s + 0.200·25-s − 1.62·26-s − 0.192·27-s + 0.935·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.802647372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.802647372\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.63T + 2T^{2} \) |
| 13 | \( 1 + 3.14T + 13T^{2} \) |
| 17 | \( 1 - 0.677T + 17T^{2} \) |
| 19 | \( 1 - 1.95T + 19T^{2} \) |
| 23 | \( 1 - 0.293T + 23T^{2} \) |
| 29 | \( 1 + 1.82T + 29T^{2} \) |
| 31 | \( 1 + 1.86T + 31T^{2} \) |
| 37 | \( 1 + 4.03T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 2.89T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 7.35T + 59T^{2} \) |
| 61 | \( 1 + 7.22T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 + 5.77T + 71T^{2} \) |
| 73 | \( 1 - 9.27T + 73T^{2} \) |
| 79 | \( 1 + 5.40T + 79T^{2} \) |
| 83 | \( 1 + 2.97T + 83T^{2} \) |
| 89 | \( 1 - 7.47T + 89T^{2} \) |
| 97 | \( 1 + 2.80T + 97T^{2} \) |
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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
−10.22482721878478764287449744090, −9.083733697518999035024603027769, −7.59265727181722956852424523409, −7.01911264736484179170888684938, −6.08299076107448577976659239887, −5.40565502374712576390095813362, −4.76983769111103109061002971676, −3.85799005662132767044643755673, −2.72937683253695944941053018271, −1.63118793542702921115555211920,
1.63118793542702921115555211920, 2.72937683253695944941053018271, 3.85799005662132767044643755673, 4.76983769111103109061002971676, 5.40565502374712576390095813362, 6.08299076107448577976659239887, 7.01911264736484179170888684938, 7.59265727181722956852424523409, 9.083733697518999035024603027769, 10.22482721878478764287449744090
