L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.292 − 0.707i)5-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (0.292 + 0.707i)10-s − 2i·13-s − 1.00·16-s + (−0.707 − 0.707i)17-s − 1.00·18-s + (−0.707 − 0.292i)20-s + (0.292 + 0.292i)25-s + (1.41 + 1.41i)26-s + (0.707 − 1.70i)29-s + (0.707 − 0.707i)32-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.292 − 0.707i)5-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (0.292 + 0.707i)10-s − 2i·13-s − 1.00·16-s + (−0.707 − 0.707i)17-s − 1.00·18-s + (−0.707 − 0.292i)20-s + (0.292 + 0.292i)25-s + (1.41 + 1.41i)26-s + (0.707 − 1.70i)29-s + (0.707 − 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8862506743\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8862506743\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + 2iT - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.589208867553381184203361238168, −7.921759785439069081325943956238, −7.50828873845244617560292049602, −6.51769060372961369775851950822, −5.73311025379446444825846240177, −5.00870124223874830053303421204, −4.51237447014751953929728692299, −2.95616701517561944424574615945, −1.84739879112385287718000702336, −0.70358521393780651185275790086,
1.46124516365071390515862013025, 2.15841504260812958125026092373, 3.28716366349171627962390798530, 4.02956970599213676033029826720, 4.76587169000204867696532546397, 6.28704985212214390620944585903, 6.94693701199420381132638685521, 7.12166805901827682209405060849, 8.560957037447836926655235083050, 8.905747559005528762401293988171