L(s) = 1 | − 3-s + 2·7-s + 9-s − 11-s − 4·13-s − 2·17-s − 2·21-s + 2·23-s − 5·25-s − 27-s + 2·29-s + 4·31-s + 33-s − 6·37-s + 4·39-s − 6·41-s − 12·43-s + 6·47-s − 3·49-s + 2·51-s + 2·63-s + 4·67-s − 2·69-s + 10·71-s + 2·73-s + 5·75-s − 2·77-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.485·17-s − 0.436·21-s + 0.417·23-s − 25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.174·33-s − 0.986·37-s + 0.640·39-s − 0.937·41-s − 1.82·43-s + 0.875·47-s − 3/7·49-s + 0.280·51-s + 0.251·63-s + 0.488·67-s − 0.240·69-s + 1.18·71-s + 0.234·73-s + 0.577·75-s − 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.588292958367558716534570992774, −7.942710865660897618647461812074, −7.09459379389830635630745773189, −6.42993764417006864713469932537, −5.24274058652077590095165528950, −4.93912381390666761861872331878, −3.90529526587309291887022370964, −2.61580326498876244659774387394, −1.58005593402553417518764847925, 0,
1.58005593402553417518764847925, 2.61580326498876244659774387394, 3.90529526587309291887022370964, 4.93912381390666761861872331878, 5.24274058652077590095165528950, 6.42993764417006864713469932537, 7.09459379389830635630745773189, 7.942710865660897618647461812074, 8.588292958367558716534570992774