L(s) = 1 | + 2-s − 1.33·3-s + 4-s − 5-s − 1.33·6-s + 1.60·7-s + 8-s − 1.22·9-s − 10-s + 3.78·11-s − 1.33·12-s + 13-s + 1.60·14-s + 1.33·15-s + 16-s − 7.50·17-s − 1.22·18-s + 1.98·19-s − 20-s − 2.14·21-s + 3.78·22-s − 7.33·23-s − 1.33·24-s + 25-s + 26-s + 5.62·27-s + 1.60·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.769·3-s + 0.5·4-s − 0.447·5-s − 0.544·6-s + 0.607·7-s + 0.353·8-s − 0.408·9-s − 0.316·10-s + 1.14·11-s − 0.384·12-s + 0.277·13-s + 0.429·14-s + 0.344·15-s + 0.250·16-s − 1.82·17-s − 0.288·18-s + 0.455·19-s − 0.223·20-s − 0.467·21-s + 0.806·22-s − 1.53·23-s − 0.272·24-s + 0.200·25-s + 0.196·26-s + 1.08·27-s + 0.303·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + 1.33T + 3T^{2} \) |
| 7 | \( 1 - 1.60T + 7T^{2} \) |
| 11 | \( 1 - 3.78T + 11T^{2} \) |
| 17 | \( 1 + 7.50T + 17T^{2} \) |
| 19 | \( 1 - 1.98T + 19T^{2} \) |
| 23 | \( 1 + 7.33T + 23T^{2} \) |
| 29 | \( 1 + 9.25T + 29T^{2} \) |
| 37 | \( 1 + 2.62T + 37T^{2} \) |
| 41 | \( 1 + 6.29T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 - 0.235T + 47T^{2} \) |
| 53 | \( 1 - 5.87T + 53T^{2} \) |
| 59 | \( 1 - 0.614T + 59T^{2} \) |
| 61 | \( 1 - 6.39T + 61T^{2} \) |
| 67 | \( 1 + 0.858T + 67T^{2} \) |
| 71 | \( 1 + 2.99T + 71T^{2} \) |
| 73 | \( 1 + 0.697T + 73T^{2} \) |
| 79 | \( 1 + 9.61T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 6.48T + 89T^{2} \) |
| 97 | \( 1 - 0.423T + 97T^{2} \) |
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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
−7.999651280016826414565105530525, −7.09971949864646189844603325614, −6.48320623784207862424729961668, −5.79234112080214066926734029566, −5.12823526264123995728848708536, −4.15744713456309670116596082155, −3.83445539348733219093480712173, −2.47755763776379686511947491664, −1.50322302499238233980405843474, 0,
1.50322302499238233980405843474, 2.47755763776379686511947491664, 3.83445539348733219093480712173, 4.15744713456309670116596082155, 5.12823526264123995728848708536, 5.79234112080214066926734029566, 6.48320623784207862424729961668, 7.09971949864646189844603325614, 7.999651280016826414565105530525