L(s) = 1 | + 2-s − 0.339·3-s + 4-s − 2.83·5-s − 0.339·6-s − 3.26·7-s + 8-s − 2.88·9-s − 2.83·10-s − 2.84·11-s − 0.339·12-s − 5.10·13-s − 3.26·14-s + 0.962·15-s + 16-s − 6.71·17-s − 2.88·18-s − 0.852·19-s − 2.83·20-s + 1.10·21-s − 2.84·22-s − 23-s − 0.339·24-s + 3.02·25-s − 5.10·26-s + 1.99·27-s − 3.26·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.196·3-s + 0.5·4-s − 1.26·5-s − 0.138·6-s − 1.23·7-s + 0.353·8-s − 0.961·9-s − 0.895·10-s − 0.859·11-s − 0.0981·12-s − 1.41·13-s − 0.871·14-s + 0.248·15-s + 0.250·16-s − 1.62·17-s − 0.679·18-s − 0.195·19-s − 0.633·20-s + 0.241·21-s − 0.607·22-s − 0.208·23-s − 0.0693·24-s + 0.605·25-s − 1.00·26-s + 0.384·27-s − 0.616·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1197602315\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1197602315\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 + 0.339T + 3T^{2} \) |
| 5 | \( 1 + 2.83T + 5T^{2} \) |
| 7 | \( 1 + 3.26T + 7T^{2} \) |
| 11 | \( 1 + 2.84T + 11T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 + 6.71T + 17T^{2} \) |
| 19 | \( 1 + 0.852T + 19T^{2} \) |
| 29 | \( 1 + 5.13T + 29T^{2} \) |
| 31 | \( 1 - 9.14T + 31T^{2} \) |
| 37 | \( 1 - 3.06T + 37T^{2} \) |
| 41 | \( 1 + 7.10T + 41T^{2} \) |
| 43 | \( 1 + 1.19T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 0.435T + 53T^{2} \) |
| 59 | \( 1 + 1.02T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 5.64T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 + 9.82T + 73T^{2} \) |
| 79 | \( 1 - 4.13T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 + 6.78T + 89T^{2} \) |
| 97 | \( 1 + 9.73T + 97T^{2} \) |
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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
−8.033245671180085986622342282891, −7.18987275379461788269290637373, −6.66331981572211079864546262664, −5.97360403165288430355697899750, −5.02028610742301011555475159024, −4.52096027717935749869045680488, −3.62255965195479430406685148981, −2.88746526295183226035534060605, −2.32608997015657904788889586503, −0.14814336621678904802362426607,
0.14814336621678904802362426607, 2.32608997015657904788889586503, 2.88746526295183226035534060605, 3.62255965195479430406685148981, 4.52096027717935749869045680488, 5.02028610742301011555475159024, 5.97360403165288430355697899750, 6.66331981572211079864546262664, 7.18987275379461788269290637373, 8.033245671180085986622342282891