| L(s) = 1 | + (−0.841 + 0.540i)2-s + (0.425 − 2.96i)3-s + (0.415 − 0.909i)4-s + (−1.34 + 0.393i)5-s + (1.24 + 2.72i)6-s + (2.81 + 3.24i)7-s + (0.142 + 0.989i)8-s + (−5.71 − 1.67i)9-s + (0.915 − 1.05i)10-s + (0.186 + 0.119i)11-s + (−2.51 − 1.61i)12-s + (−1.30 + 1.50i)13-s + (−4.11 − 1.20i)14-s + (0.595 + 4.14i)15-s + (−0.654 − 0.755i)16-s + (0.0462 + 0.101i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 + 0.382i)2-s + (0.245 − 1.70i)3-s + (0.207 − 0.454i)4-s + (−0.599 + 0.176i)5-s + (0.507 + 1.11i)6-s + (1.06 + 1.22i)7-s + (0.0503 + 0.349i)8-s + (−1.90 − 0.559i)9-s + (0.289 − 0.334i)10-s + (0.0561 + 0.0361i)11-s + (−0.726 − 0.466i)12-s + (−0.361 + 0.416i)13-s + (−1.10 − 0.323i)14-s + (0.153 + 1.06i)15-s + (−0.163 − 0.188i)16-s + (0.0112 + 0.0245i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.620677 - 0.214487i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.620677 - 0.214487i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (0.936 - 4.70i)T \) |
| good | 3 | \( 1 + (-0.425 + 2.96i)T + (-2.87 - 0.845i)T^{2} \) |
| 5 | \( 1 + (1.34 - 0.393i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-2.81 - 3.24i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.186 - 0.119i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (1.30 - 1.50i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.0462 - 0.101i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.34 + 2.94i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (1.48 + 3.24i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.876 + 6.09i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (4.05 + 1.19i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (3.27 - 0.961i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.462 + 3.21i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 7.73T + 47T^{2} \) |
| 53 | \( 1 + (0.0956 + 0.110i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-8.62 + 9.95i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.809 - 5.63i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (12.0 - 7.75i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (4.91 - 3.16i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (1.57 - 3.44i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-5.71 + 6.59i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-15.0 - 4.40i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.145 - 1.00i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-17.4 + 5.13i)T + (81.6 - 52.4i)T^{2} \) |
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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
−15.51419462634301439449632686170, −14.64198464653821606587924675968, −13.46151665838264432940941397056, −11.92875510753389318902316372822, −11.50707328610745818355163951379, −9.076617530803137308492866526731, −8.009318793067340437571198750101, −7.21899669341582847478469587947, −5.68029211454484492482606442953, −2.08715073228369810181640206899,
3.69290666418932711763248501216, 4.80415990622208720577708199194, 7.70132028244341484742038563584, 8.761120507345938525616114815085, 10.24211814927421798550903607132, 10.72847839238668409430807706204, 11.96611092531565747526864843024, 14.02397523034210410638795029506, 14.94975116629302034838359856202, 16.14230946628321807328090645073
