L(s) = 1 | + (1 − i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + 2.44·6-s + (0.758 − 0.758i)7-s + (−2 − 2i)8-s + 2.99i·9-s − 14.9·11-s + (2.44 − 2.44i)12-s + (−16.9 − 16.9i)13-s − 1.51i·14-s − 4·16-s + (−17.9 + 17.9i)17-s + (2.99 + 2.99i)18-s − 12.3i·19-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + 0.408·6-s + (0.108 − 0.108i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s − 1.36·11-s + (0.204 − 0.204i)12-s + (−1.30 − 1.30i)13-s − 0.108i·14-s − 0.250·16-s + (−1.05 + 1.05i)17-s + (0.166 + 0.166i)18-s − 0.647i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6550460985\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6550460985\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.758 + 0.758i)T - 49iT^{2} \) |
| 11 | \( 1 + 14.9T + 121T^{2} \) |
| 13 | \( 1 + (16.9 + 16.9i)T + 169iT^{2} \) |
| 17 | \( 1 + (17.9 - 17.9i)T - 289iT^{2} \) |
| 19 | \( 1 + 12.3iT - 361T^{2} \) |
| 23 | \( 1 + (-9.70 - 9.70i)T + 529iT^{2} \) |
| 29 | \( 1 + 17.2iT - 841T^{2} \) |
| 31 | \( 1 + 51.7T + 961T^{2} \) |
| 37 | \( 1 + (-32.5 + 32.5i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 41.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + (14.2 + 14.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-31.5 + 31.5i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (13.0 + 13.0i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 19.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 109.T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-30.3 + 30.3i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 117.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-27.7 - 27.7i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 112. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-79.5 - 79.5i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 27.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-30.9 + 30.9i)T - 9.40e3iT^{2} \) |
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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
−9.876730025306043546820542112029, −9.062611592631673073474565908406, −7.946993263558417591999475726027, −7.30029437350301807846369234077, −5.79209042939222232841929971233, −5.08173495047689161522923963489, −4.16483981985689385511434053076, −2.94393026847117158079646847306, −2.20548867207006600192210496891, −0.16023209070921260359455673410,
2.10617845894724682844386128323, 2.93418916812215228838062975542, 4.43626014716889432549828752598, 5.09337880787294439641039656524, 6.27532849753985930354569038698, 7.28324033723865407779057414405, 7.63713277066187097489266704977, 8.850104073646201162303345298131, 9.446163176952311632418160958527, 10.64322229811314964887054827264