L(s) = 1 | + 2.51·2-s − i·3-s + 4.34·4-s + 1.21·5-s − 2.51i·6-s + (−2.34 − 1.21i)7-s + 5.89·8-s − 9-s + 3.06·10-s + 1.13i·11-s − 4.34i·12-s + 0.518i·13-s + (−5.91 − 3.06i)14-s − 1.21i·15-s + 6.16·16-s + 3.06·17-s + ⋯ |
L(s) = 1 | + 1.78·2-s − 0.577i·3-s + 2.17·4-s + 0.544·5-s − 1.02i·6-s + (−0.887 − 0.460i)7-s + 2.08·8-s − 0.333·9-s + 0.969·10-s + 0.341i·11-s − 1.25i·12-s + 0.143i·13-s + (−1.58 − 0.819i)14-s − 0.314i·15-s + 1.54·16-s + 0.743·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.65520 - 1.02999i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.65520 - 1.02999i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 + (2.34 + 1.21i)T \) |
| 23 | \( 1 + (0.341 - 4.78i)T \) |
good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 5 | \( 1 - 1.21T + 5T^{2} \) |
| 11 | \( 1 - 1.13iT - 11T^{2} \) |
| 13 | \( 1 - 0.518iT - 13T^{2} \) |
| 17 | \( 1 - 3.06T + 17T^{2} \) |
| 19 | \( 1 + 1.13T + 19T^{2} \) |
| 29 | \( 1 + 2.17T + 29T^{2} \) |
| 31 | \( 1 - 6.85iT - 31T^{2} \) |
| 37 | \( 1 + 10.1iT - 37T^{2} \) |
| 41 | \( 1 + 5iT - 41T^{2} \) |
| 43 | \( 1 - 6.71iT - 43T^{2} \) |
| 47 | \( 1 - 4.21iT - 47T^{2} \) |
| 53 | \( 1 + 5.41iT - 53T^{2} \) |
| 59 | \( 1 + 0.200iT - 59T^{2} \) |
| 61 | \( 1 - 6.21T + 61T^{2} \) |
| 67 | \( 1 - 3.65iT - 67T^{2} \) |
| 71 | \( 1 - 2.16T + 71T^{2} \) |
| 73 | \( 1 + 10.2iT - 73T^{2} \) |
| 79 | \( 1 + 8.59iT - 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 5.11T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−11.25311627916326923230305655012, −10.27145409471775384132910112617, −9.282025370806009358442365436949, −7.62175249550353624293404368319, −6.91319184898036515896029692914, −6.03402793939408395297871360294, −5.35988328992556087287975614079, −4.02251910027282078791975478779, −3.11429098855797465091356591436, −1.85124753333109692068992102188,
2.38302198315059345400285523448, 3.31548534409300025987463814489, 4.27563264790569603267693407158, 5.45700309157031209395557555337, 5.98097677126360064282372885401, 6.83912482200847545459477871977, 8.289588766715559944389545572105, 9.591325463204760199212353796854, 10.29493881416988199806862375946, 11.36321449581920535379284004141