| L(s) = 1 | − 2-s − 1.79·3-s + 4-s − 0.791·5-s + 1.79·6-s − 2·7-s − 8-s + 0.208·9-s + 0.791·10-s + 0.791·11-s − 1.79·12-s + 3.79·13-s + 2·14-s + 1.41·15-s + 16-s − 7.58·17-s − 0.208·18-s − 1.58·19-s − 0.791·20-s + 3.58·21-s − 0.791·22-s + 3.79·23-s + 1.79·24-s − 4.37·25-s − 3.79·26-s + 5.00·27-s − 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.03·3-s + 0.5·4-s − 0.353·5-s + 0.731·6-s − 0.755·7-s − 0.353·8-s + 0.0695·9-s + 0.250·10-s + 0.238·11-s − 0.517·12-s + 1.05·13-s + 0.534·14-s + 0.365·15-s + 0.250·16-s − 1.83·17-s − 0.0491·18-s − 0.363·19-s − 0.176·20-s + 0.781·21-s − 0.168·22-s + 0.790·23-s + 0.365·24-s − 0.874·25-s − 0.743·26-s + 0.962·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3754634198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3754634198\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + 1.79T + 3T^{2} \) |
| 5 | \( 1 + 0.791T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 0.791T + 11T^{2} \) |
| 13 | \( 1 - 3.79T + 13T^{2} \) |
| 17 | \( 1 + 7.58T + 17T^{2} \) |
| 19 | \( 1 + 1.58T + 19T^{2} \) |
| 23 | \( 1 - 3.79T + 23T^{2} \) |
| 29 | \( 1 - 3.79T + 29T^{2} \) |
| 31 | \( 1 + 8.37T + 31T^{2} \) |
| 41 | \( 1 + 9.79T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 + 1.58T + 53T^{2} \) |
| 59 | \( 1 + 1.58T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 6.37T + 67T^{2} \) |
| 71 | \( 1 + 9.16T + 71T^{2} \) |
| 73 | \( 1 - 4.37T + 73T^{2} \) |
| 79 | \( 1 + 8.20T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−8.837642282287718538101310215467, −8.256365932034525382611161759387, −7.08571739450326183426519248766, −6.53529017629075219936491099299, −6.05172819376431863356360216262, −5.06156410192951241488903686949, −4.04398449495997574497365445529, −3.11086782906438425603096515517, −1.82456115682906265388772580004, −0.43375729369958759628604280137,
0.43375729369958759628604280137, 1.82456115682906265388772580004, 3.11086782906438425603096515517, 4.04398449495997574497365445529, 5.06156410192951241488903686949, 6.05172819376431863356360216262, 6.53529017629075219936491099299, 7.08571739450326183426519248766, 8.256365932034525382611161759387, 8.837642282287718538101310215467
