L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s + 25·5-s − 36·6-s + 33·7-s + 64·8-s + 81·9-s + 100·10-s + 39·11-s − 144·12-s − 314·13-s + 132·14-s − 225·15-s + 256·16-s − 1.94e3·17-s + 324·18-s + 922·19-s + 400·20-s − 297·21-s + 156·22-s + 3.84e3·23-s − 576·24-s + 625·25-s − 1.25e3·26-s − 729·27-s + 528·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.254·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.0971·11-s − 0.288·12-s − 0.515·13-s + 0.179·14-s − 0.258·15-s + 1/4·16-s − 1.63·17-s + 0.235·18-s + 0.585·19-s + 0.223·20-s − 0.146·21-s + 0.0687·22-s + 1.51·23-s − 0.204·24-s + 1/5·25-s − 0.364·26-s − 0.192·27-s + 0.127·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| 37 | \( 1 + p^{2} T \) |
good | 7 | \( 1 - 33 T + p^{5} T^{2} \) |
| 11 | \( 1 - 39 T + p^{5} T^{2} \) |
| 13 | \( 1 + 314 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1949 T + p^{5} T^{2} \) |
| 19 | \( 1 - 922 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3844 T + p^{5} T^{2} \) |
| 29 | \( 1 + 8313 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5985 T + p^{5} T^{2} \) |
| 41 | \( 1 + 13607 T + p^{5} T^{2} \) |
| 43 | \( 1 - 847 T + p^{5} T^{2} \) |
| 47 | \( 1 - 2904 T + p^{5} T^{2} \) |
| 53 | \( 1 + 33851 T + p^{5} T^{2} \) |
| 59 | \( 1 + 2186 T + p^{5} T^{2} \) |
| 61 | \( 1 - 19893 T + p^{5} T^{2} \) |
| 67 | \( 1 + 18596 T + p^{5} T^{2} \) |
| 71 | \( 1 + 17740 T + p^{5} T^{2} \) |
| 73 | \( 1 + 44536 T + p^{5} T^{2} \) |
| 79 | \( 1 - 79732 T + p^{5} T^{2} \) |
| 83 | \( 1 - 36254 T + p^{5} T^{2} \) |
| 89 | \( 1 + 57970 T + p^{5} T^{2} \) |
| 97 | \( 1 - 85531 T + p^{5} T^{2} \) |
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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
−8.817625449677982462190695239139, −7.58972333863139846140215361084, −6.82287394271354722154467743141, −6.14266795133016546948277594002, −5.07916334086974064967327647066, −4.69365901533234397167874136417, −3.44822094613482913873296857996, −2.33648225453125840007258238095, −1.37313653918855264984020667891, 0,
1.37313653918855264984020667891, 2.33648225453125840007258238095, 3.44822094613482913873296857996, 4.69365901533234397167874136417, 5.07916334086974064967327647066, 6.14266795133016546948277594002, 6.82287394271354722154467743141, 7.58972333863139846140215361084, 8.817625449677982462190695239139