L(s) = 1 | + 3-s − 2·7-s + 9-s + 4·11-s − 13-s − 6·19-s − 2·21-s + 6·23-s + 27-s − 8·29-s + 4·33-s − 10·37-s − 39-s − 10·41-s + 12·43-s + 12·47-s − 3·49-s + 6·53-s − 6·57-s + 12·59-s − 10·61-s − 2·63-s + 4·67-s + 6·69-s − 8·71-s + 4·73-s − 8·77-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 1.37·19-s − 0.436·21-s + 1.25·23-s + 0.192·27-s − 1.48·29-s + 0.696·33-s − 1.64·37-s − 0.160·39-s − 1.56·41-s + 1.82·43-s + 1.75·47-s − 3/7·49-s + 0.824·53-s − 0.794·57-s + 1.56·59-s − 1.28·61-s − 0.251·63-s + 0.488·67-s + 0.722·69-s − 0.949·71-s + 0.468·73-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−15.28095865238890, −14.85353376096094, −14.31016012143633, −13.85438597415370, −13.08869830472415, −12.90047334149188, −12.21439176918458, −11.75879842578572, −10.99066756665578, −10.50318838051784, −9.950815304139681, −9.235885036069966, −8.868263318338495, −8.614860347317702, −7.515648162112321, −7.166108472756470, −6.598442126060236, −6.044033499826265, −5.292640948821192, −4.533237975986246, −3.759532972298341, −3.545800991506883, −2.573010789020403, −1.972639566682210, −1.094085955791147, 0,
1.094085955791147, 1.972639566682210, 2.573010789020403, 3.545800991506883, 3.759532972298341, 4.533237975986246, 5.292640948821192, 6.044033499826265, 6.598442126060236, 7.166108472756470, 7.515648162112321, 8.614860347317702, 8.868263318338495, 9.235885036069966, 9.950815304139681, 10.50318838051784, 10.99066756665578, 11.75879842578572, 12.21439176918458, 12.90047334149188, 13.08869830472415, 13.85438597415370, 14.31016012143633, 14.85353376096094, 15.28095865238890