L(s) = 1 | + (−0.382 + 0.923i)3-s − i·4-s + (0.923 − 0.382i)7-s + (−0.707 − 0.707i)9-s + (0.923 + 0.382i)12-s − i·13-s − 16-s + (0.707 − 0.707i)19-s + i·21-s + (0.707 + 0.707i)25-s + (0.923 − 0.382i)27-s + (−0.382 − 0.923i)28-s + (−0.382 + 0.923i)31-s + (−0.707 + 0.707i)36-s + (0.382 − 0.923i)37-s + ⋯ |
L(s) = 1 | + (−0.382 + 0.923i)3-s − i·4-s + (0.923 − 0.382i)7-s + (−0.707 − 0.707i)9-s + (0.923 + 0.382i)12-s − i·13-s − 16-s + (0.707 − 0.707i)19-s + i·21-s + (0.707 + 0.707i)25-s + (0.923 − 0.382i)27-s + (−0.382 − 0.923i)28-s + (−0.382 + 0.923i)31-s + (−0.707 + 0.707i)36-s + (0.382 − 0.923i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9314219448\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9314219448\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.382 - 0.923i)T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + iT^{2} \) |
| 5 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.923 + 0.382i)T + (0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + iT - T^{2} \) |
| 19 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 23 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (0.382 - 0.923i)T + (-0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (-0.382 + 0.923i)T + (-0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (0.923 - 0.382i)T + (0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (1.84 + 0.765i)T + (0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.765 - 1.84i)T + (-0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.923 + 0.382i)T + (0.707 + 0.707i)T^{2} \) |
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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
−10.51702431749081360994305337261, −9.546737502633754113643737465979, −8.926504317292487008967179000453, −7.79333506575039778910056053382, −6.74449893247520555049894623690, −5.57593247734728820978415414316, −5.13625372891182705930577463548, −4.25526848313584173900644069948, −2.91060173063213558025377020512, −1.13058756060716069541281489473,
1.70664793217847276202914391771, 2.75223936571192429260620120560, 4.16727839334714006367345887990, 5.14741104467599233897211220774, 6.24303958371950751958844354577, 7.12745520646295418411273826227, 7.87969406980296998016776681781, 8.462689093889677136667488750344, 9.342662452827607580036227253391, 10.71575720278223634646088388502