| L(s) = 1 | − 2·2-s − 9.63·3-s + 4·4-s − 2.86·5-s + 19.2·6-s + 7·7-s − 8·8-s + 65.8·9-s + 5.72·10-s − 45.8·11-s − 38.5·12-s − 14·14-s + 27.5·15-s + 16·16-s + 11.4·17-s − 131.·18-s + 61.8·19-s − 11.4·20-s − 67.4·21-s + 91.6·22-s − 205.·23-s + 77.0·24-s − 116.·25-s − 374.·27-s + 28·28-s + 294.·29-s − 55.1·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.85·3-s + 0.5·4-s − 0.255·5-s + 1.31·6-s + 0.377·7-s − 0.353·8-s + 2.43·9-s + 0.180·10-s − 1.25·11-s − 0.927·12-s − 0.267·14-s + 0.474·15-s + 0.250·16-s + 0.163·17-s − 1.72·18-s + 0.746·19-s − 0.127·20-s − 0.700·21-s + 0.888·22-s − 1.86·23-s + 0.655·24-s − 0.934·25-s − 2.66·27-s + 0.188·28-s + 1.88·29-s − 0.335·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4174231904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4174231904\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 7 | \( 1 - 7T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 9.63T + 27T^{2} \) |
| 5 | \( 1 + 2.86T + 125T^{2} \) |
| 11 | \( 1 + 45.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 11.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 61.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 205.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 294.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 244.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 374.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 15.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 184.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 505.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 174.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 22.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 466.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 574.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 81.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 213.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 948.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 270.T + 9.12e5T^{2} \) |
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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
−8.404692854984963723021172355047, −7.79958020512428248969058132546, −7.11597657833698438249048778598, −6.19975980033059464136961292753, −5.63751521907719623213638110918, −4.88321692944341227531736738748, −4.03223151738665914239567556540, −2.51969320845533305058008267974, −1.31701319478620311074714571982, −0.38857755905529916203730209909,
0.38857755905529916203730209909, 1.31701319478620311074714571982, 2.51969320845533305058008267974, 4.03223151738665914239567556540, 4.88321692944341227531736738748, 5.63751521907719623213638110918, 6.19975980033059464136961292753, 7.11597657833698438249048778598, 7.79958020512428248969058132546, 8.404692854984963723021172355047
