L(s) = 1 | + (−88.2 + 88.2i)2-s + 2.08e6i·4-s + (1.67e7 + 1.40e7i)5-s + (6.37e8 + 6.37e8i)7-s + (−3.68e8 − 3.68e8i)8-s + (−2.71e9 + 2.42e8i)10-s − 1.46e11i·11-s + (−1.71e11 + 1.71e11i)13-s − 1.12e11·14-s − 4.30e12·16-s + (5.82e12 − 5.82e12i)17-s + 2.23e13i·19-s + (−2.91e13 + 3.48e13i)20-s + (1.29e13 + 1.29e13i)22-s + (2.49e14 + 2.49e14i)23-s + ⋯ |
L(s) = 1 | + (−0.0609 + 0.0609i)2-s + 0.992i·4-s + (0.767 + 0.641i)5-s + (0.853 + 0.853i)7-s + (−0.121 − 0.121i)8-s + (−0.0858 + 0.00767i)10-s − 1.70i·11-s + (−0.344 + 0.344i)13-s − 0.104·14-s − 0.977·16-s + (0.700 − 0.700i)17-s + 0.835i·19-s + (−0.636 + 0.761i)20-s + (0.104 + 0.104i)22-s + (1.25 + 1.25i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(2.653793787\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.653793787\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.67e7 - 1.40e7i)T \) |
good | 2 | \( 1 + (88.2 - 88.2i)T - 2.09e6iT^{2} \) |
| 7 | \( 1 + (-6.37e8 - 6.37e8i)T + 5.58e17iT^{2} \) |
| 11 | \( 1 + 1.46e11iT - 7.40e21T^{2} \) |
| 13 | \( 1 + (1.71e11 - 1.71e11i)T - 2.47e23iT^{2} \) |
| 17 | \( 1 + (-5.82e12 + 5.82e12i)T - 6.90e25iT^{2} \) |
| 19 | \( 1 - 2.23e13iT - 7.14e26T^{2} \) |
| 23 | \( 1 + (-2.49e14 - 2.49e14i)T + 3.94e28iT^{2} \) |
| 29 | \( 1 + 2.46e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 6.47e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + (-3.36e16 - 3.36e16i)T + 8.55e32iT^{2} \) |
| 41 | \( 1 + 6.87e16iT - 7.38e33T^{2} \) |
| 43 | \( 1 + (1.55e17 - 1.55e17i)T - 2.00e34iT^{2} \) |
| 47 | \( 1 + (-5.27e16 + 5.27e16i)T - 1.30e35iT^{2} \) |
| 53 | \( 1 + (-1.53e17 - 1.53e17i)T + 1.62e36iT^{2} \) |
| 59 | \( 1 + 5.42e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 6.13e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + (-1.80e19 - 1.80e19i)T + 2.22e38iT^{2} \) |
| 71 | \( 1 - 2.47e19iT - 7.52e38T^{2} \) |
| 73 | \( 1 + (8.19e18 - 8.19e18i)T - 1.34e39iT^{2} \) |
| 79 | \( 1 + 1.12e19iT - 7.08e39T^{2} \) |
| 83 | \( 1 + (-6.21e19 - 6.21e19i)T + 1.99e40iT^{2} \) |
| 89 | \( 1 - 3.66e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + (-6.32e19 - 6.32e19i)T + 5.27e41iT^{2} \) |
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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
−11.77244038461193120915011814768, −11.22778473994480772234657062206, −9.576822320623108374386801983008, −8.531923126593003161500719055799, −7.55239953453189538733386852745, −6.16181506204129394171712976218, −5.13717564268146393560930440441, −3.37007886265582003451406192536, −2.65063792302329717677692594403, −1.29513731123489126125512448377,
0.55720603223889480793093386240, 1.41874210645177708049879432038, 2.29121245689482727838676087994, 4.56905861558606992971300001194, 4.98864247595751286493305270954, 6.42524994482635073741317675640, 7.65190132775092076913695844966, 9.159089525141175216180058022965, 10.07561165772763710803916464148, 10.85073134526262861154457055011