L(s) = 1 | + 0.618·2-s − 3.23·3-s − 1.61·4-s − 5-s − 2.00·6-s − 2.38·7-s − 2.23·8-s + 7.47·9-s − 0.618·10-s − 11-s + 5.23·12-s − 0.381·13-s − 1.47·14-s + 3.23·15-s + 1.85·16-s + 6.09·17-s + 4.61·18-s − 5·19-s + 1.61·20-s + 7.70·21-s − 0.618·22-s − 4.38·23-s + 7.23·24-s − 4·25-s − 0.236·26-s − 14.4·27-s + 3.85·28-s + ⋯ |
L(s) = 1 | + 0.437·2-s − 1.86·3-s − 0.809·4-s − 0.447·5-s − 0.816·6-s − 0.900·7-s − 0.790·8-s + 2.49·9-s − 0.195·10-s − 0.301·11-s + 1.51·12-s − 0.105·13-s − 0.393·14-s + 0.835·15-s + 0.463·16-s + 1.47·17-s + 1.08·18-s − 1.14·19-s + 0.361·20-s + 1.68·21-s − 0.131·22-s − 0.913·23-s + 1.47·24-s − 0.800·25-s − 0.0462·26-s − 2.78·27-s + 0.728·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 4027 | \( 1+O(T) \) |
good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 2.38T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 0.381T + 13T^{2} \) |
| 17 | \( 1 - 6.09T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 4.38T + 23T^{2} \) |
| 29 | \( 1 + 5.61T + 29T^{2} \) |
| 31 | \( 1 + 8.23T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 7.23T + 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 + 8.47T + 47T^{2} \) |
| 53 | \( 1 - 6.23T + 53T^{2} \) |
| 59 | \( 1 + 8.70T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 7.70T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 - 3.56T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−7.50939616460320270353633694237, −6.72054655645144530211915249311, −5.99677309543797091783548848235, −5.45100587679577749404641465168, −4.92909907810301224659728900900, −3.84247413746456909930358062807, −3.56164333894819981168153762588, −1.66990175773918597488610417692, 0, 0,
1.66990175773918597488610417692, 3.56164333894819981168153762588, 3.84247413746456909930358062807, 4.92909907810301224659728900900, 5.45100587679577749404641465168, 5.99677309543797091783548848235, 6.72054655645144530211915249311, 7.50939616460320270353633694237