L(s) = 1 | + 3-s + 5-s + 4·7-s − 2·9-s − 3·11-s + 13-s + 15-s − 6·17-s − 5·19-s + 4·21-s + 3·23-s + 25-s − 5·27-s − 3·29-s − 7·31-s − 3·33-s + 4·35-s − 2·37-s + 39-s + 2·43-s − 2·45-s − 10·47-s + 9·49-s − 6·51-s − 2·53-s − 3·55-s − 5·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s − 2/3·9-s − 0.904·11-s + 0.277·13-s + 0.258·15-s − 1.45·17-s − 1.14·19-s + 0.872·21-s + 0.625·23-s + 1/5·25-s − 0.962·27-s − 0.557·29-s − 1.25·31-s − 0.522·33-s + 0.676·35-s − 0.328·37-s + 0.160·39-s + 0.304·43-s − 0.298·45-s − 1.45·47-s + 9/7·49-s − 0.840·51-s − 0.274·53-s − 0.404·55-s − 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 151 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.86425536650892942039848442499, −7.17294583806648994699114595000, −6.25065519215269131482577573003, −5.46917463712188149340929953001, −4.86169920150271159075951413138, −4.13854227006604527930966691520, −3.05265541852131274170127356576, −2.18327748408234942978567680215, −1.71519011409642791184043151116, 0,
1.71519011409642791184043151116, 2.18327748408234942978567680215, 3.05265541852131274170127356576, 4.13854227006604527930966691520, 4.86169920150271159075951413138, 5.46917463712188149340929953001, 6.25065519215269131482577573003, 7.17294583806648994699114595000, 7.86425536650892942039848442499