L(s) = 1 | + i·2-s + (1.61 − 0.618i)3-s − 4-s + (0.618 + 1.61i)6-s + (−2.61 + 0.381i)7-s − i·8-s + (2.23 − 2.00i)9-s − 4.47i·11-s + (−1.61 + 0.618i)12-s − 1.23i·13-s + (−0.381 − 2.61i)14-s + 16-s − 5.23·17-s + (2.00 + 2.23i)18-s − 8.47i·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.934 − 0.356i)3-s − 0.5·4-s + (0.252 + 0.660i)6-s + (−0.989 + 0.144i)7-s − 0.353i·8-s + (0.745 − 0.666i)9-s − 1.34i·11-s + (−0.467 + 0.178i)12-s − 0.342i·13-s + (−0.102 − 0.699i)14-s + 0.250·16-s − 1.26·17-s + (0.471 + 0.527i)18-s − 1.94i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.485239903\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.485239903\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-1.61 + 0.618i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.61 - 0.381i)T \) |
good | 11 | \( 1 + 4.47iT - 11T^{2} \) |
| 13 | \( 1 + 1.23iT - 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 + 8.47iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 7.70iT - 29T^{2} \) |
| 31 | \( 1 - 2.76iT - 31T^{2} \) |
| 37 | \( 1 + 0.763T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 - 4.94T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 - 0.472iT - 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 7.23iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 7.23iT - 71T^{2} \) |
| 73 | \( 1 + 11.2iT - 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 5.52T + 89T^{2} \) |
| 97 | \( 1 - 0.763iT - 97T^{2} \) |
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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
−9.300508467250497076700932491945, −8.839834757276002089325093229443, −8.328535453250362704579344269396, −6.91967579864884439934010747571, −6.81783940051827124334884273252, −5.68556478748470897119337239977, −4.48036056640841011429084416583, −3.33776836665511901497326538238, −2.59889782380549954575825303729, −0.58126049432332938940186507270,
1.80268033401864132652597302012, 2.63752327217385364092547780433, 3.95604944487530147394437667590, 4.20956438804436174170426597315, 5.66133238956742418883971715771, 6.87171882124857471827774169151, 7.69117245278843988481040447618, 8.617202945280402515977981238404, 9.598556776476758320000296098527, 9.805372156430722560620490400632