L(s) = 1 | + (−0.707 − 0.707i)3-s + (1.86 + 1.24i)5-s + 1.58·7-s + 1.00i·9-s + (−3.92 − 3.92i)11-s + (−3.10 − 3.10i)13-s + (−0.438 − 2.19i)15-s + 1.48i·17-s + (−4.94 + 4.94i)19-s + (−1.12 − 1.12i)21-s − 6.61·23-s + (1.92 + 4.61i)25-s + (0.707 − 0.707i)27-s + (−4.42 + 4.42i)29-s − 1.50·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (0.831 + 0.554i)5-s + 0.600·7-s + 0.333i·9-s + (−1.18 − 1.18i)11-s + (−0.861 − 0.861i)13-s + (−0.113 − 0.566i)15-s + 0.359i·17-s + (−1.13 + 1.13i)19-s + (−0.245 − 0.245i)21-s − 1.38·23-s + (0.384 + 0.923i)25-s + (0.136 − 0.136i)27-s + (−0.821 + 0.821i)29-s − 0.270·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.02155817458\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02155817458\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.86 - 1.24i)T \) |
good | 7 | \( 1 - 1.58T + 7T^{2} \) |
| 11 | \( 1 + (3.92 + 3.92i)T + 11iT^{2} \) |
| 13 | \( 1 + (3.10 + 3.10i)T + 13iT^{2} \) |
| 17 | \( 1 - 1.48iT - 17T^{2} \) |
| 19 | \( 1 + (4.94 - 4.94i)T - 19iT^{2} \) |
| 23 | \( 1 + 6.61T + 23T^{2} \) |
| 29 | \( 1 + (4.42 - 4.42i)T - 29iT^{2} \) |
| 31 | \( 1 + 1.50T + 31T^{2} \) |
| 37 | \( 1 + (-2.14 + 2.14i)T - 37iT^{2} \) |
| 41 | \( 1 - 6.84iT - 41T^{2} \) |
| 43 | \( 1 + (0.322 - 0.322i)T - 43iT^{2} \) |
| 47 | \( 1 + 13.3iT - 47T^{2} \) |
| 53 | \( 1 + (-0.931 + 0.931i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.14 + 1.14i)T + 59iT^{2} \) |
| 61 | \( 1 + (2.67 - 2.67i)T - 61iT^{2} \) |
| 67 | \( 1 + (-5.43 - 5.43i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.26iT - 71T^{2} \) |
| 73 | \( 1 + 5.27T + 73T^{2} \) |
| 79 | \( 1 + 6.52T + 79T^{2} \) |
| 83 | \( 1 + (-0.973 - 0.973i)T + 83iT^{2} \) |
| 89 | \( 1 + 6.83iT - 89T^{2} \) |
| 97 | \( 1 - 15.8iT - 97T^{2} \) |
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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
−9.831374294491056412585529845459, −8.499226752789904826128680165127, −8.026038979526565021239437113434, −7.26305188671616716720100471798, −6.14481467345381906886144677247, −5.70701050528201436207067169882, −5.02122401649354877979181467111, −3.63124991818907248242360468273, −2.54469115094219869728429747648, −1.71978416036638269167592425737,
0.00728936746541235961878949609, 1.90767626579082202783602406044, 2.46579509188478973097098816232, 4.38951821074248405229849372343, 4.64914424679174887706724762472, 5.47120929962213502170308960706, 6.35039869859680951580121691030, 7.29774468509159001341508095266, 8.044576259733265961171146408887, 9.082170994196532891589289825813