L(s) = 1 | − 1.92·2-s − 3-s + 1.71·4-s + 5-s + 1.92·6-s + 2.81·7-s + 0.555·8-s + 9-s − 1.92·10-s + 6.22·11-s − 1.71·12-s − 1.67·13-s − 5.41·14-s − 15-s − 4.49·16-s − 3.11·17-s − 1.92·18-s + 3.95·19-s + 1.71·20-s − 2.81·21-s − 11.9·22-s − 4.92·23-s − 0.555·24-s + 25-s + 3.23·26-s − 27-s + 4.81·28-s + ⋯ |
L(s) = 1 | − 1.36·2-s − 0.577·3-s + 0.855·4-s + 0.447·5-s + 0.786·6-s + 1.06·7-s + 0.196·8-s + 0.333·9-s − 0.609·10-s + 1.87·11-s − 0.494·12-s − 0.465·13-s − 1.44·14-s − 0.258·15-s − 1.12·16-s − 0.755·17-s − 0.454·18-s + 0.907·19-s + 0.382·20-s − 0.613·21-s − 2.55·22-s − 1.02·23-s − 0.113·24-s + 0.200·25-s + 0.634·26-s − 0.192·27-s + 0.909·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.097432045\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.097432045\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 + 1.92T + 2T^{2} \) |
| 7 | \( 1 - 2.81T + 7T^{2} \) |
| 11 | \( 1 - 6.22T + 11T^{2} \) |
| 13 | \( 1 + 1.67T + 13T^{2} \) |
| 17 | \( 1 + 3.11T + 17T^{2} \) |
| 19 | \( 1 - 3.95T + 19T^{2} \) |
| 23 | \( 1 + 4.92T + 23T^{2} \) |
| 29 | \( 1 - 3.08T + 29T^{2} \) |
| 31 | \( 1 - 3.99T + 31T^{2} \) |
| 37 | \( 1 + 6.31T + 37T^{2} \) |
| 41 | \( 1 + 9.19T + 41T^{2} \) |
| 43 | \( 1 - 3.08T + 43T^{2} \) |
| 47 | \( 1 - 1.23T + 47T^{2} \) |
| 53 | \( 1 - 3.04T + 53T^{2} \) |
| 59 | \( 1 - 8.57T + 59T^{2} \) |
| 61 | \( 1 - 4.21T + 61T^{2} \) |
| 67 | \( 1 - 7.61T + 67T^{2} \) |
| 71 | \( 1 - 6.98T + 71T^{2} \) |
| 73 | \( 1 + 6.17T + 73T^{2} \) |
| 79 | \( 1 + 2.98T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 0.243T + 89T^{2} \) |
| 97 | \( 1 - 3.34T + 97T^{2} \) |
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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
−8.345756709842171491323850042399, −7.38510105950278445577550259056, −6.82595331163845885354398499580, −6.24142633982615216174700546330, −5.18505250102913702162225610184, −4.57485600178325505610339561773, −3.74304524002808570041624686392, −2.18099933580092498155235831785, −1.55506322210489343149449251636, −0.76388122952817668872201853882,
0.76388122952817668872201853882, 1.55506322210489343149449251636, 2.18099933580092498155235831785, 3.74304524002808570041624686392, 4.57485600178325505610339561773, 5.18505250102913702162225610184, 6.24142633982615216174700546330, 6.82595331163845885354398499580, 7.38510105950278445577550259056, 8.345756709842171491323850042399