L(s) = 1 | + (1.86 + 1.07i)3-s + (−0.817 − 2.08i)5-s + (2.54 − 1.46i)7-s + (0.817 + 1.41i)9-s + (−0.317 + 0.550i)11-s + (3.60 − 0.0716i)13-s + (0.716 − 4.76i)15-s + (−1.05 + 0.611i)17-s + (−0.682 − 1.18i)19-s + 6.32·21-s + (−1.86 − 1.07i)23-s + (−3.66 + 3.40i)25-s − 2.93i·27-s + (1.5 − 2.59i)29-s + 8.96·31-s + ⋯ |
L(s) = 1 | + (1.07 + 0.621i)3-s + (−0.365 − 0.930i)5-s + (0.961 − 0.555i)7-s + (0.272 + 0.472i)9-s + (−0.0957 + 0.165i)11-s + (0.999 − 0.0198i)13-s + (0.184 − 1.22i)15-s + (−0.257 + 0.148i)17-s + (−0.156 − 0.271i)19-s + 1.38·21-s + (−0.388 − 0.224i)23-s + (−0.732 + 0.680i)25-s − 0.565i·27-s + (0.278 − 0.482i)29-s + 1.60·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.425900546\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.425900546\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.817 + 2.08i)T \) |
| 13 | \( 1 + (-3.60 + 0.0716i)T \) |
good | 3 | \( 1 + (-1.86 - 1.07i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-2.54 + 1.46i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.317 - 0.550i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.05 - 0.611i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.682 + 1.18i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.86 + 1.07i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8.96T + 31T^{2} \) |
| 37 | \( 1 + (1.05 + 0.611i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.98 + 8.62i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.18 - 0.683i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6.16iT - 47T^{2} \) |
| 53 | \( 1 + 0.642iT - 53T^{2} \) |
| 59 | \( 1 + (3.79 + 6.57i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.13 - 1.96i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.95 - 4.01i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.31 - 2.28i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 10.3iT - 73T^{2} \) |
| 79 | \( 1 - 1.03T + 79T^{2} \) |
| 83 | \( 1 - 11.8iT - 83T^{2} \) |
| 89 | \( 1 + (6.27 - 10.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (12.8 - 7.39i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.682327226104143294031229525016, −8.934151394713604442889973708854, −8.178228696679199024133733716153, −7.940434099913182340201915056906, −6.55155093182416885937986654989, −5.28445095865193036377979268123, −4.26583182780408302095595982062, −3.93542889419720329488326880805, −2.50926166875816843744106645503, −1.11414172672724402846744736124,
1.59190360034883034003677036905, 2.60095601349640785186624101007, 3.41341640649135322335512125935, 4.59227076241103073550690799376, 5.90322614476755604027719743935, 6.75685174091565718361770040412, 7.72461681843562940934518897629, 8.263303299093143636986917424214, 8.778691458454208604494096654353, 9.953647688347304413841770635319