L(s) = 1 | − 2.68·2-s + 1.14·3-s + 5.19·4-s + 0.981·5-s − 3.07·6-s − 5.03·7-s − 8.56·8-s − 1.68·9-s − 2.63·10-s + 11-s + 5.95·12-s − 2.90·13-s + 13.4·14-s + 1.12·15-s + 12.5·16-s − 17-s + 4.52·18-s − 0.273·19-s + 5.09·20-s − 5.76·21-s − 2.68·22-s − 0.344·23-s − 9.81·24-s − 4.03·25-s + 7.78·26-s − 5.37·27-s − 26.1·28-s + ⋯ |
L(s) = 1 | − 1.89·2-s + 0.661·3-s + 2.59·4-s + 0.438·5-s − 1.25·6-s − 1.90·7-s − 3.02·8-s − 0.562·9-s − 0.832·10-s + 0.301·11-s + 1.71·12-s − 0.804·13-s + 3.60·14-s + 0.290·15-s + 3.14·16-s − 0.242·17-s + 1.06·18-s − 0.0626·19-s + 1.13·20-s − 1.25·21-s − 0.571·22-s − 0.0718·23-s − 2.00·24-s − 0.807·25-s + 1.52·26-s − 1.03·27-s − 4.93·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2432050582\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2432050582\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 + 2.68T + 2T^{2} \) |
| 3 | \( 1 - 1.14T + 3T^{2} \) |
| 5 | \( 1 - 0.981T + 5T^{2} \) |
| 7 | \( 1 + 5.03T + 7T^{2} \) |
| 13 | \( 1 + 2.90T + 13T^{2} \) |
| 19 | \( 1 + 0.273T + 19T^{2} \) |
| 23 | \( 1 + 0.344T + 23T^{2} \) |
| 29 | \( 1 + 5.98T + 29T^{2} \) |
| 31 | \( 1 - 0.228T + 31T^{2} \) |
| 37 | \( 1 - 0.957T + 37T^{2} \) |
| 41 | \( 1 + 8.93T + 41T^{2} \) |
| 47 | \( 1 + 5.63T + 47T^{2} \) |
| 53 | \( 1 - 0.243T + 53T^{2} \) |
| 59 | \( 1 - 6.15T + 59T^{2} \) |
| 61 | \( 1 - 3.10T + 61T^{2} \) |
| 67 | \( 1 + 9.16T + 67T^{2} \) |
| 71 | \( 1 + 0.195T + 71T^{2} \) |
| 73 | \( 1 - 0.0243T + 73T^{2} \) |
| 79 | \( 1 - 0.00321T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 1.74T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.963094341737339539780366048260, −7.29508266437659645822194289110, −6.69929012152055874516738202228, −6.14858164698465434712943566774, −5.47926202062918693830255331813, −3.78733314746983147738655255187, −3.08939733358674058999780300554, −2.47344728202480790602972256015, −1.74999280714537142156166507107, −0.29199868872548448701200508897,
0.29199868872548448701200508897, 1.74999280714537142156166507107, 2.47344728202480790602972256015, 3.08939733358674058999780300554, 3.78733314746983147738655255187, 5.47926202062918693830255331813, 6.14858164698465434712943566774, 6.69929012152055874516738202228, 7.29508266437659645822194289110, 7.963094341737339539780366048260