L(s) = 1 | + (0.707 − 1.58i)3-s + 1.41i·5-s − i·7-s + (−2.00 − 2.23i)9-s + 4.47·11-s − 0.837·13-s + (2.23 + 1.00i)15-s − 1.64i·17-s − 7.16i·19-s + (−1.58 − 0.707i)21-s + 5.65·23-s + 2.99·25-s + (−4.94 + 1.58i)27-s − 7.30i·29-s + 6.32i·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.912i)3-s + 0.632i·5-s − 0.377i·7-s + (−0.666 − 0.745i)9-s + 1.34·11-s − 0.232·13-s + (0.577 + 0.258i)15-s − 0.398i·17-s − 1.64i·19-s + (−0.345 − 0.154i)21-s + 1.17·23-s + 0.599·25-s + (−0.952 + 0.304i)27-s − 1.35i·29-s + 1.13i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43951 - 0.991107i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43951 - 0.991107i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 1.58i)T \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 + 0.837T + 13T^{2} \) |
| 17 | \( 1 + 1.64iT - 17T^{2} \) |
| 19 | \( 1 + 7.16iT - 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 7.30iT - 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 + 8.32T + 37T^{2} \) |
| 41 | \( 1 - 1.18iT - 41T^{2} \) |
| 43 | \( 1 + 4.32iT - 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 10.1iT - 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 3.16T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 1.18T + 71T^{2} \) |
| 73 | \( 1 + 8.32T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 10.1iT - 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−10.43169577550765251398237465414, −9.128863025577550589625023657202, −8.822952262189913924124742479556, −7.32608449502901790593179772176, −7.02158452246868418936553174074, −6.21666417321056354989205606381, −4.77848505323463176883509018189, −3.46065635417491917337146806182, −2.51725997783230688746864151443, −1.00223227001158987024373141102,
1.63114332403665162694278117867, 3.24175488480860312762508301777, 4.12690368845679156928238890074, 5.09922228460727550506679782017, 5.99339349753668320810140090517, 7.24068265789125074127477034451, 8.499647663025469116498515056363, 8.883690308461253352283844103832, 9.694621618537934619245263573856, 10.53251811413264089749985918418