L(s) = 1 | + 1.59·2-s − 3-s + 0.556·4-s + 1.72·5-s − 1.59·6-s − 4.76·7-s − 2.30·8-s + 9-s + 2.75·10-s + 5.76·11-s − 0.556·12-s + 6.38·13-s − 7.61·14-s − 1.72·15-s − 4.80·16-s + 17-s + 1.59·18-s + 6.07·19-s + 0.960·20-s + 4.76·21-s + 9.22·22-s − 1.53·23-s + 2.30·24-s − 2.02·25-s + 10.2·26-s − 27-s − 2.65·28-s + ⋯ |
L(s) = 1 | + 1.13·2-s − 0.577·3-s + 0.278·4-s + 0.771·5-s − 0.652·6-s − 1.80·7-s − 0.815·8-s + 0.333·9-s + 0.872·10-s + 1.73·11-s − 0.160·12-s + 1.77·13-s − 2.03·14-s − 0.445·15-s − 1.20·16-s + 0.242·17-s + 0.376·18-s + 1.39·19-s + 0.214·20-s + 1.03·21-s + 1.96·22-s − 0.319·23-s + 0.471·24-s − 0.404·25-s + 2.00·26-s − 0.192·27-s − 0.501·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.943239872\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.943239872\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 - 1.59T + 2T^{2} \) |
| 5 | \( 1 - 1.72T + 5T^{2} \) |
| 7 | \( 1 + 4.76T + 7T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 13 | \( 1 - 6.38T + 13T^{2} \) |
| 19 | \( 1 - 6.07T + 19T^{2} \) |
| 23 | \( 1 + 1.53T + 23T^{2} \) |
| 29 | \( 1 + 6.74T + 29T^{2} \) |
| 31 | \( 1 + 0.347T + 31T^{2} \) |
| 37 | \( 1 + 6.53T + 37T^{2} \) |
| 41 | \( 1 + 2.27T + 41T^{2} \) |
| 43 | \( 1 - 9.57T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 + 8.79T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 3.48T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 5.97T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 5.71T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 19.2T + 97T^{2} \) |
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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
−7.47923794878221352237025032530, −6.63122399577918845738837122166, −6.13585987590837004076051578259, −5.96682385026335272973365434613, −5.23942560246295634858855837606, −4.04286321677361434337760051848, −3.63128833932810259982701191890, −3.18706465224327939118051439022, −1.80252923579406511135557233335, −0.75550228911694147217461530547,
0.75550228911694147217461530547, 1.80252923579406511135557233335, 3.18706465224327939118051439022, 3.63128833932810259982701191890, 4.04286321677361434337760051848, 5.23942560246295634858855837606, 5.96682385026335272973365434613, 6.13585987590837004076051578259, 6.63122399577918845738837122166, 7.47923794878221352237025032530