L(s) = 1 | − i·2-s − i·3-s − 4-s + (−2.14 + 0.627i)5-s − 6-s − 0.255i·7-s + i·8-s − 9-s + (0.627 + 2.14i)10-s + 2.03·11-s + i·12-s + 1.70i·13-s − 0.255·14-s + (0.627 + 2.14i)15-s + 16-s + 5.83i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.959 + 0.280i)5-s − 0.408·6-s − 0.0964i·7-s + 0.353i·8-s − 0.333·9-s + (0.198 + 0.678i)10-s + 0.614·11-s + 0.288i·12-s + 0.473i·13-s − 0.0681·14-s + (0.162 + 0.554i)15-s + 0.250·16-s + 1.41i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.896119 + 0.128326i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.896119 + 0.128326i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 + (2.14 - 0.627i)T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + 0.255iT - 7T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 - 1.70iT - 13T^{2} \) |
| 17 | \( 1 - 5.83iT - 17T^{2} \) |
| 19 | \( 1 + 6.09T + 19T^{2} \) |
| 23 | \( 1 - 4.83iT - 23T^{2} \) |
| 29 | \( 1 - 2.58T + 29T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 0.255iT - 43T^{2} \) |
| 47 | \( 1 - 9.83iT - 47T^{2} \) |
| 53 | \( 1 - 1.16iT - 53T^{2} \) |
| 59 | \( 1 - 5.09T + 59T^{2} \) |
| 61 | \( 1 - 0.160T + 61T^{2} \) |
| 67 | \( 1 + 7.38iT - 67T^{2} \) |
| 71 | \( 1 + 5.64T + 71T^{2} \) |
| 73 | \( 1 - 9.93iT - 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 8.42iT - 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 4.87iT - 97T^{2} \) |
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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
−10.36019751932403737769035405333, −9.191139221037191315633892515740, −8.444677496671700801275640934962, −7.71017603000176916106542868770, −6.72957701907450987536875014463, −5.90795553461681228344939407726, −4.34704890597374620454112435075, −3.83717696812735753892781644990, −2.55562327694977164531846327964, −1.28640420678961987446190023845,
0.47980034374181565795987252548, 2.79782609462110289642465101269, 4.07560655381895995282978745950, 4.59957708624467858620712586220, 5.63967621378665196005227340396, 6.70628609147252988705699630840, 7.47829180347776677973065330056, 8.537164115714719925499328096909, 8.860520925298505985271384666271, 9.930563479378253778052293690262