L(s) = 1 | + 1.79·2-s + 1.79·3-s + 1.20·4-s − 0.791·5-s + 3.20·6-s + 7-s − 1.41·8-s + 0.208·9-s − 1.41·10-s + 3.79·11-s + 2.16·12-s + 13-s + 1.79·14-s − 1.41·15-s − 4.95·16-s + 3.79·17-s + 0.373·18-s − 8.58·19-s − 0.956·20-s + 1.79·21-s + 6.79·22-s − 23-s − 2.53·24-s − 4.37·25-s + 1.79·26-s − 5.00·27-s + 1.20·28-s + ⋯ |
L(s) = 1 | + 1.26·2-s + 1.03·3-s + 0.604·4-s − 0.353·5-s + 1.30·6-s + 0.377·7-s − 0.501·8-s + 0.0695·9-s − 0.448·10-s + 1.14·11-s + 0.625·12-s + 0.277·13-s + 0.478·14-s − 0.365·15-s − 1.23·16-s + 0.919·17-s + 0.0881·18-s − 1.96·19-s − 0.213·20-s + 0.390·21-s + 1.44·22-s − 0.208·23-s − 0.518·24-s − 0.874·25-s + 0.351·26-s − 0.962·27-s + 0.228·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.884016956\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.884016956\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 1.79T + 2T^{2} \) |
| 3 | \( 1 - 1.79T + 3T^{2} \) |
| 5 | \( 1 + 0.791T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 3.79T + 11T^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 19 | \( 1 + 8.58T + 19T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 - 7.79T + 31T^{2} \) |
| 37 | \( 1 - 2.41T + 37T^{2} \) |
| 41 | \( 1 + 8.58T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 0.791T + 47T^{2} \) |
| 53 | \( 1 - 7T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 - 1.41T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 0.791T + 79T^{2} \) |
| 83 | \( 1 + 9.95T + 83T^{2} \) |
| 89 | \( 1 - 2.41T + 89T^{2} \) |
| 97 | \( 1 - 4.58T + 97T^{2} \) |
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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
−11.92505265902164863697108272449, −11.23203715128744075153853487667, −9.732593514825492897127083296102, −8.738184254063746517951580972828, −8.041366207182376171893546134447, −6.64200893389648589734657156833, −5.63124658457004451210042106662, −4.16570582004888027511718622988, −3.66588514228427334282968642564, −2.25186174094201216200891282432,
2.25186174094201216200891282432, 3.66588514228427334282968642564, 4.16570582004888027511718622988, 5.63124658457004451210042106662, 6.64200893389648589734657156833, 8.041366207182376171893546134447, 8.738184254063746517951580972828, 9.732593514825492897127083296102, 11.23203715128744075153853487667, 11.92505265902164863697108272449