L(s) = 1 | + 3-s − 3·7-s − 2·9-s + 2·11-s − 13-s − 5·17-s + 19-s − 3·21-s + 23-s − 5·25-s − 5·27-s + 3·29-s − 4·31-s + 2·33-s − 2·37-s − 39-s − 8·41-s − 8·43-s + 8·47-s + 2·49-s − 5·51-s − 9·53-s + 57-s + 59-s − 14·61-s + 6·63-s + 13·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.13·7-s − 2/3·9-s + 0.603·11-s − 0.277·13-s − 1.21·17-s + 0.229·19-s − 0.654·21-s + 0.208·23-s − 25-s − 0.962·27-s + 0.557·29-s − 0.718·31-s + 0.348·33-s − 0.328·37-s − 0.160·39-s − 1.24·41-s − 1.21·43-s + 1.16·47-s + 2/7·49-s − 0.700·51-s − 1.23·53-s + 0.132·57-s + 0.130·59-s − 1.79·61-s + 0.755·63-s + 1.58·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.253041138673007375271029562828, −8.699258028013535951563233483351, −7.74515606034087698025852781244, −6.73715959969007181198215210739, −6.17043864352631516446211866530, −5.03277316009204668285879882866, −3.81156163044135505245852369486, −3.09698588528238924660584466755, −2.01374174404058579498750722555, 0,
2.01374174404058579498750722555, 3.09698588528238924660584466755, 3.81156163044135505245852369486, 5.03277316009204668285879882866, 6.17043864352631516446211866530, 6.73715959969007181198215210739, 7.74515606034087698025852781244, 8.699258028013535951563233483351, 9.253041138673007375271029562828