L(s) = 1 | + 2-s + 2.23·3-s + 4-s − 4.44·5-s + 2.23·6-s + 4.19·7-s + 8-s + 1.98·9-s − 4.44·10-s + 0.346·11-s + 2.23·12-s − 4.24·13-s + 4.19·14-s − 9.92·15-s + 16-s − 6.81·17-s + 1.98·18-s + 19-s − 4.44·20-s + 9.35·21-s + 0.346·22-s + 0.0232·23-s + 2.23·24-s + 14.7·25-s − 4.24·26-s − 2.27·27-s + 4.19·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.28·3-s + 0.5·4-s − 1.98·5-s + 0.911·6-s + 1.58·7-s + 0.353·8-s + 0.660·9-s − 1.40·10-s + 0.104·11-s + 0.644·12-s − 1.17·13-s + 1.12·14-s − 2.56·15-s + 0.250·16-s − 1.65·17-s + 0.467·18-s + 0.229·19-s − 0.994·20-s + 2.04·21-s + 0.0738·22-s + 0.00484·23-s + 0.455·24-s + 2.95·25-s − 0.832·26-s − 0.437·27-s + 0.792·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.022909135\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.022909135\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 + 4.44T + 5T^{2} \) |
| 7 | \( 1 - 4.19T + 7T^{2} \) |
| 11 | \( 1 - 0.346T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 6.81T + 17T^{2} \) |
| 23 | \( 1 - 0.0232T + 23T^{2} \) |
| 29 | \( 1 + 0.890T + 29T^{2} \) |
| 31 | \( 1 - 5.80T + 31T^{2} \) |
| 37 | \( 1 - 8.53T + 37T^{2} \) |
| 41 | \( 1 + 1.37T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 9.01T + 47T^{2} \) |
| 53 | \( 1 - 4.13T + 53T^{2} \) |
| 59 | \( 1 - 1.14T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 3.99T + 71T^{2} \) |
| 73 | \( 1 - 8.95T + 73T^{2} \) |
| 79 | \( 1 + 8.22T + 79T^{2} \) |
| 83 | \( 1 + 4.88T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−7.971965588693580670172956850789, −7.27001648851493106889992996468, −6.86305010049763806400410145537, −5.43792971139989284020650495450, −4.56434199149217914255668474473, −4.31419537387395848042976287394, −3.71943175278866609150799148231, −2.58885296957416716028995060182, −2.32268880473538647374542148577, −0.828834796380139286460982441025,
0.828834796380139286460982441025, 2.32268880473538647374542148577, 2.58885296957416716028995060182, 3.71943175278866609150799148231, 4.31419537387395848042976287394, 4.56434199149217914255668474473, 5.43792971139989284020650495450, 6.86305010049763806400410145537, 7.27001648851493106889992996468, 7.971965588693580670172956850789