L(s) = 1 | + 5-s + 2·13-s + 2·17-s + 8·23-s + 25-s − 2·29-s − 8·31-s + 6·37-s − 2·41-s − 43-s − 7·49-s + 2·53-s − 8·59-s + 10·61-s + 2·65-s + 4·67-s + 14·73-s + 8·79-s + 4·83-s + 2·85-s + 6·89-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.554·13-s + 0.485·17-s + 1.66·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.986·37-s − 0.312·41-s − 0.152·43-s − 49-s + 0.274·53-s − 1.04·59-s + 1.28·61-s + 0.248·65-s + 0.488·67-s + 1.63·73-s + 0.900·79-s + 0.439·83-s + 0.216·85-s + 0.635·89-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.69217848849151, −13.16224999306791, −13.00982682169319, −12.37357818876289, −11.89629077276483, −11.13342935209167, −10.96314428029259, −10.54158706098255, −9.636219352232957, −9.511857573010341, −8.994125463873185, −8.360262844345682, −7.909204669476026, −7.324240441263454, −6.744996475844436, −6.373135354191950, −5.688223124731976, −5.166668390914354, −4.883129620921704, −3.848281973989864, −3.628455388601521, −2.824152863611623, −2.313671854690495, −1.449157049574382, −1.034910641136110, 0,
1.034910641136110, 1.449157049574382, 2.313671854690495, 2.824152863611623, 3.628455388601521, 3.848281973989864, 4.883129620921704, 5.166668390914354, 5.688223124731976, 6.373135354191950, 6.744996475844436, 7.324240441263454, 7.909204669476026, 8.360262844345682, 8.994125463873185, 9.511857573010341, 9.636219352232957, 10.54158706098255, 10.96314428029259, 11.13342935209167, 11.89629077276483, 12.37357818876289, 13.00982682169319, 13.16224999306791, 13.69217848849151