L(s) = 1 | − 2.49·2-s − 1.66·3-s + 4.22·4-s + 4.15·6-s + 4.24·7-s − 5.56·8-s − 0.229·9-s − 6.39·11-s − 7.03·12-s + 1.34·13-s − 10.6·14-s + 5.42·16-s − 1.79·17-s + 0.573·18-s − 3.80·19-s − 7.07·21-s + 15.9·22-s − 5.33·23-s + 9.25·24-s − 3.34·26-s + 5.37·27-s + 17.9·28-s + 1.62·29-s − 3.96·31-s − 2.41·32-s + 10.6·33-s + 4.48·34-s + ⋯ |
L(s) = 1 | − 1.76·2-s − 0.960·3-s + 2.11·4-s + 1.69·6-s + 1.60·7-s − 1.96·8-s − 0.0766·9-s − 1.92·11-s − 2.03·12-s + 0.371·13-s − 2.83·14-s + 1.35·16-s − 0.436·17-s + 0.135·18-s − 0.872·19-s − 1.54·21-s + 3.40·22-s − 1.11·23-s + 1.88·24-s − 0.656·26-s + 1.03·27-s + 3.39·28-s + 0.301·29-s − 0.712·31-s − 0.426·32-s + 1.85·33-s + 0.769·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 3 | \( 1 + 1.66T + 3T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 + 6.39T + 11T^{2} \) |
| 13 | \( 1 - 1.34T + 13T^{2} \) |
| 17 | \( 1 + 1.79T + 17T^{2} \) |
| 19 | \( 1 + 3.80T + 19T^{2} \) |
| 23 | \( 1 + 5.33T + 23T^{2} \) |
| 29 | \( 1 - 1.62T + 29T^{2} \) |
| 31 | \( 1 + 3.96T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 2.25T + 41T^{2} \) |
| 43 | \( 1 + 1.94T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 9.06T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 4.44T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 - 4.44T + 73T^{2} \) |
| 79 | \( 1 + 9.81T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 5.74T + 97T^{2} \) |
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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
−8.010633689420518582796559407286, −7.35437158097137586012310123029, −6.41937803754891748273879599143, −5.71689834686185362601927700903, −5.09071327365035182435219059616, −4.23682581585466986138495776304, −2.54217302048696002789529142978, −2.10401590680767151242198010325, −0.938972289586566936880348466428, 0,
0.938972289586566936880348466428, 2.10401590680767151242198010325, 2.54217302048696002789529142978, 4.23682581585466986138495776304, 5.09071327365035182435219059616, 5.71689834686185362601927700903, 6.41937803754891748273879599143, 7.35437158097137586012310123029, 8.010633689420518582796559407286