L(s) = 1 | + 2.41·2-s + 3-s + 3.82·4-s − 5-s + 2.41·6-s − 1.41·7-s + 4.41·8-s + 9-s − 2.41·10-s − 2.24·11-s + 3.82·12-s − 3.41·13-s − 3.41·14-s − 15-s + 2.99·16-s + 1.17·17-s + 2.41·18-s − 19-s − 3.82·20-s − 1.41·21-s − 5.41·22-s + 7.65·23-s + 4.41·24-s + 25-s − 8.24·26-s + 27-s − 5.41·28-s + ⋯ |
L(s) = 1 | + 1.70·2-s + 0.577·3-s + 1.91·4-s − 0.447·5-s + 0.985·6-s − 0.534·7-s + 1.56·8-s + 0.333·9-s − 0.763·10-s − 0.676·11-s + 1.10·12-s − 0.946·13-s − 0.912·14-s − 0.258·15-s + 0.749·16-s + 0.284·17-s + 0.569·18-s − 0.229·19-s − 0.856·20-s − 0.308·21-s − 1.15·22-s + 1.59·23-s + 0.901·24-s + 0.200·25-s − 1.61·26-s + 0.192·27-s − 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.216419366\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.216419366\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 + 0.242T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 7.65T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 7.31T + 61T^{2} \) |
| 67 | \( 1 + 9.65T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 5.89T + 89T^{2} \) |
| 97 | \( 1 + 9.75T + 97T^{2} \) |
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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
−12.23511320145276539552922414293, −11.21411063461569254226128164985, −10.19067810920104136365419417778, −8.927516763719285408940852117938, −7.54003299218301302356581633802, −6.83378106786382026747386148724, −5.49238882760750975616135169751, −4.56926424233502251010085086314, −3.40287857323995405053371516534, −2.53627796200111631637271098706,
2.53627796200111631637271098706, 3.40287857323995405053371516534, 4.56926424233502251010085086314, 5.49238882760750975616135169751, 6.83378106786382026747386148724, 7.54003299218301302356581633802, 8.927516763719285408940852117938, 10.19067810920104136365419417778, 11.21411063461569254226128164985, 12.23511320145276539552922414293