L(s) = 1 | + 1.56·3-s − 5-s − 7-s − 0.561·9-s + 1.56·11-s − 3.56·13-s − 1.56·15-s + 0.438·17-s − 1.56·21-s + 25-s − 5.56·27-s − 6.68·29-s − 3.12·31-s + 2.43·33-s + 35-s + 1.12·37-s − 5.56·39-s + 2·41-s − 4·43-s + 0.561·45-s + 0.684·47-s + 49-s + 0.684·51-s − 13.1·53-s − 1.56·55-s − 6.24·59-s + 2·61-s + ⋯ |
L(s) = 1 | + 0.901·3-s − 0.447·5-s − 0.377·7-s − 0.187·9-s + 0.470·11-s − 0.987·13-s − 0.403·15-s + 0.106·17-s − 0.340·21-s + 0.200·25-s − 1.07·27-s − 1.24·29-s − 0.560·31-s + 0.424·33-s + 0.169·35-s + 0.184·37-s − 0.890·39-s + 0.312·41-s − 0.609·43-s + 0.0837·45-s + 0.0998·47-s + 0.142·49-s + 0.0958·51-s − 1.80·53-s − 0.210·55-s − 0.813·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 - 0.438T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 3.12T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 0.684T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 6.24T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 0.246T + 73T^{2} \) |
| 79 | \( 1 - 7.80T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−8.739258981859685727783109207165, −7.79767775479679517618969943550, −7.38475579020328726436737389070, −6.39141597342121333975559988429, −5.46572023579994268774300016196, −4.44152135326070623920419020624, −3.54418853156303576582044155516, −2.86009336980223243862642897105, −1.79870447151663116465203390608, 0,
1.79870447151663116465203390608, 2.86009336980223243862642897105, 3.54418853156303576582044155516, 4.44152135326070623920419020624, 5.46572023579994268774300016196, 6.39141597342121333975559988429, 7.38475579020328726436737389070, 7.79767775479679517618969943550, 8.739258981859685727783109207165