L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − 4i·7-s − i·8-s − 9-s + i·12-s + 4i·13-s + 4·14-s + 16-s + 6i·17-s − i·18-s − 19-s − 4·21-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.51i·7-s − 0.353i·8-s − 0.333·9-s + 0.288i·12-s + 1.10i·13-s + 1.06·14-s + 0.250·16-s + 1.45i·17-s − 0.235i·18-s − 0.229·19-s − 0.872·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.380077582\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.380077582\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.762601137633688930135565974823, −7.84790887721582945729195550272, −7.45388836060171032454165352073, −6.69157433898784711523042244955, −6.13319976906138436544438566732, −5.15144791684935465233929567914, −4.01701106225434364186231577418, −3.74572846576651904593028088118, −2.01538314346297996997146711218, −1.00450375836979626562020635740,
0.53151822188718079537270376493, 2.29966579180626510198145790575, 2.75440449176235264017235152792, 3.71737891447654124832833402850, 4.83656939869977604559892487213, 5.36276075021088364313858603462, 6.06219415243581732653210482748, 7.21488981173211428289124788735, 8.320189834051550421184466012893, 8.707924546066177169490290949439