L(s) = 1 | + (−1 − 1.73i)5-s + 2·13-s + (−3 + 5.19i)17-s + (−2 − 3.46i)19-s + (−2 − 3.46i)23-s + (0.500 − 0.866i)25-s − 6·29-s + (−4 + 6.92i)31-s + (5 + 8.66i)37-s − 10·41-s + 12·43-s + (4 + 6.92i)47-s + (3 − 5.19i)53-s + (−2 + 3.46i)59-s + (−5 − 8.66i)61-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.774i)5-s + 0.554·13-s + (−0.727 + 1.26i)17-s + (−0.458 − 0.794i)19-s + (−0.417 − 0.722i)23-s + (0.100 − 0.173i)25-s − 1.11·29-s + (−0.718 + 1.24i)31-s + (0.821 + 1.42i)37-s − 1.56·41-s + 1.82·43-s + (0.583 + 1.01i)47-s + (0.412 − 0.713i)53-s + (−0.260 + 0.450i)59-s + (−0.640 − 1.10i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6333828253\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6333828253\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.684550184011888664313147393708, −8.271751151504484456256315239142, −7.36321481715390707455473010977, −6.48365262117789232112006071306, −5.87741474396597399751439054251, −4.78518800285213778716035309575, −4.29369951988747574389436800426, −3.42240605440209132099647609503, −2.22615830680149792870735143565, −1.15935882014592665747642053826,
0.19865364056486288441435363131, 1.81661694810478825367068465998, 2.76218814417333193748341307604, 3.73910200655359719564867049375, 4.26022168705495187138359738218, 5.54441547913117532361085288489, 6.01150472475383942470278631495, 7.21933097359971682697723695376, 7.32114585838750228813597999279, 8.287473194015390875199421027266