L(s) = 1 | + (1.41 − 1.41i)3-s + (0.707 + 0.707i)7-s − 3.00i·9-s + 2.00·21-s + (−2.82 − 2.82i)27-s + 2i·29-s + (−1.41 − 1.41i)47-s + 1.00i·49-s + (2.12 − 2.12i)63-s − 5.00·81-s + (−1.41 + 1.41i)83-s + (2.82 + 2.82i)87-s + (1.41 − 1.41i)103-s + 2i·109-s + ⋯ |
L(s) = 1 | + (1.41 − 1.41i)3-s + (0.707 + 0.707i)7-s − 3.00i·9-s + 2.00·21-s + (−2.82 − 2.82i)27-s + 2i·29-s + (−1.41 − 1.41i)47-s + 1.00i·49-s + (2.12 − 2.12i)63-s − 5.00·81-s + (−1.41 + 1.41i)83-s + (2.82 + 2.82i)87-s + (1.41 − 1.41i)103-s + 2i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.103240523\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.103240523\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - 2iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−8.605125649612419199256138984238, −8.215864155789778300892139728663, −7.36104017750431592073649656041, −6.83298328197494639637809793343, −5.96841043796301545301530426025, −4.97635516499208739163402068639, −3.69413216050077361582942058846, −2.93570284122435744531876797557, −2.05932292767583353403717760591, −1.32659965062465926214105090107,
1.79633580454938300810023398356, 2.75602755166100783356132578982, 3.63778790001779422518305702282, 4.38748499098086923225763023360, 4.81915726906227191709807959980, 5.89936702002180636902563646425, 7.24933569145496850988242972651, 7.952545329696382454953891120291, 8.338388723493983064326494027740, 9.210943462621559055696552615624