L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 2·13-s − 15-s − 19-s + 2·21-s + 25-s − 27-s − 6·29-s − 2·31-s − 2·35-s + 2·37-s − 2·39-s − 8·43-s + 45-s − 3·49-s + 6·53-s + 57-s + 6·59-s + 2·61-s − 2·63-s + 2·65-s + 4·67-s + 14·73-s − 75-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s − 0.229·19-s + 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.338·35-s + 0.328·37-s − 0.320·39-s − 1.21·43-s + 0.149·45-s − 3/7·49-s + 0.824·53-s + 0.132·57-s + 0.781·59-s + 0.256·61-s − 0.251·63-s + 0.248·65-s + 0.488·67-s + 1.63·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.980468163660774091957797449942, −6.96150126440193556937840674666, −6.55972147582409368892889392175, −5.73117378446542533964840194656, −5.23810314506767350320248259652, −4.13990939436266819798176177598, −3.44426491136301618783378659588, −2.37483811378039756632466956802, −1.31866495876643706954581701851, 0,
1.31866495876643706954581701851, 2.37483811378039756632466956802, 3.44426491136301618783378659588, 4.13990939436266819798176177598, 5.23810314506767350320248259652, 5.73117378446542533964840194656, 6.55972147582409368892889392175, 6.96150126440193556937840674666, 7.980468163660774091957797449942