L(s) = 1 | + (−0.707 + 0.707i)3-s + 4-s − 1.41i·7-s − 1.00i·9-s + (−0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s + 16-s + (−1 − i)19-s + (1.00 + 1.00i)21-s + (0.707 + 0.707i)27-s − 1.41i·28-s + (−1 + i)31-s − 1.00i·36-s + 1.41i·37-s + 1.00i·39-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s + 4-s − 1.41i·7-s − 1.00i·9-s + (−0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s + 16-s + (−1 − i)19-s + (1.00 + 1.00i)21-s + (0.707 + 0.707i)27-s − 1.41i·28-s + (−1 + i)31-s − 1.00i·36-s + 1.41i·37-s + 1.00i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.016330664\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.016330664\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
good | 2 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.41iT - T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (1 + i)T + iT^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (1 - i)T - iT^{2} \) |
| 37 | \( 1 - 1.41iT - T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 1.41T + T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 - 1.41T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + 1.41T + T^{2} \) |
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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
−10.55246638158536707821181062055, −9.637813428142605537624023915242, −8.488601605486281541362511835356, −7.44326526704159022067054932995, −6.69601581289193307198546992262, −6.02477256201757938796492405637, −4.88698430589032533651551713885, −3.93653368868660025731766528605, −3.00639951963409543631145218188, −1.16407620473532112688886912447,
1.79443879774017476328636214715, 2.40746797908336156898807806421, 3.96917881110752077191163361478, 5.59127050424141662676893264092, 5.89070990282977991581785212141, 6.71514035016707881837909317654, 7.61080779888646819442563109291, 8.459688268653417083171588633559, 9.337583858319189960235091962548, 10.65221155762070414517948967213