L(s) = 1 | − 3-s − 1.41·5-s − 2.82·7-s + 9-s + 4.24·13-s + 1.41·15-s + 4·17-s − 8·19-s + 2.82·21-s + 5.65·23-s − 2.99·25-s − 27-s + 1.41·29-s + 2.82·31-s + 4.00·35-s + 4.24·37-s − 4.24·39-s − 4·41-s − 1.41·45-s + 11.3·47-s + 1.00·49-s − 4·51-s − 7.07·53-s + 8·57-s − 12·59-s + 1.41·61-s − 2.82·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.632·5-s − 1.06·7-s + 0.333·9-s + 1.17·13-s + 0.365·15-s + 0.970·17-s − 1.83·19-s + 0.617·21-s + 1.17·23-s − 0.599·25-s − 0.192·27-s + 0.262·29-s + 0.508·31-s + 0.676·35-s + 0.697·37-s − 0.679·39-s − 0.624·41-s − 0.210·45-s + 1.65·47-s + 0.142·49-s − 0.560·51-s − 0.971·53-s + 1.05·57-s − 1.56·59-s + 0.181·61-s − 0.356·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 7.07T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 16T + 97T^{2} \) |
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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
−8.301547938317920324847061849846, −7.58814481079868803698839453668, −6.53191207803778639455269795743, −6.29994576354050127311845517312, −5.34917122388763658737063320257, −4.27499812025335440757130846422, −3.67250210351512304971810288397, −2.74390723681515814027001571393, −1.23145225157430610426849927925, 0,
1.23145225157430610426849927925, 2.74390723681515814027001571393, 3.67250210351512304971810288397, 4.27499812025335440757130846422, 5.34917122388763658737063320257, 6.29994576354050127311845517312, 6.53191207803778639455269795743, 7.58814481079868803698839453668, 8.301547938317920324847061849846