L(s) = 1 | − 3-s + 0.521·5-s + 7-s − 2·9-s − 2·11-s + 0.478·13-s − 0.521·15-s + 17-s − 2.52·19-s − 21-s + 5.88·23-s − 4.72·25-s + 5·27-s − 1.47·29-s + 6.84·31-s + 2·33-s + 0.521·35-s + 7.32·37-s − 0.478·39-s − 2.52·41-s + 2.84·43-s − 1.04·45-s − 9.88·47-s − 6·49-s − 51-s − 5.72·53-s − 1.04·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.233·5-s + 0.377·7-s − 0.666·9-s − 0.603·11-s + 0.132·13-s − 0.134·15-s + 0.242·17-s − 0.578·19-s − 0.218·21-s + 1.22·23-s − 0.945·25-s + 0.962·27-s − 0.274·29-s + 1.22·31-s + 0.348·33-s + 0.0881·35-s + 1.20·37-s − 0.0765·39-s − 0.393·41-s + 0.433·43-s − 0.155·45-s − 1.44·47-s − 0.857·49-s − 0.140·51-s − 0.786·53-s − 0.140·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 - 0.521T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 0.478T + 13T^{2} \) |
| 19 | \( 1 + 2.52T + 19T^{2} \) |
| 23 | \( 1 - 5.88T + 23T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 - 6.84T + 31T^{2} \) |
| 37 | \( 1 - 7.32T + 37T^{2} \) |
| 41 | \( 1 + 2.52T + 41T^{2} \) |
| 43 | \( 1 - 2.84T + 43T^{2} \) |
| 47 | \( 1 + 9.88T + 47T^{2} \) |
| 53 | \( 1 + 5.72T + 53T^{2} \) |
| 61 | \( 1 + 2.47T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 - 3.32T + 83T^{2} \) |
| 89 | \( 1 - 8.84T + 89T^{2} \) |
| 97 | \( 1 - 7.20T + 97T^{2} \) |
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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
−8.081751360555782217624267329596, −7.42259670650967108752949488553, −6.32985664917928411947869130406, −5.98421873904288820088493297184, −5.03594860526762005395342643400, −4.56522575124683794996913070657, −3.28741556755469027254772645352, −2.51900774308071596193943062192, −1.32881466096707980999311054514, 0,
1.32881466096707980999311054514, 2.51900774308071596193943062192, 3.28741556755469027254772645352, 4.56522575124683794996913070657, 5.03594860526762005395342643400, 5.98421873904288820088493297184, 6.32985664917928411947869130406, 7.42259670650967108752949488553, 8.081751360555782217624267329596