L(s) = 1 | + (0.433 − 0.900i)2-s + (−0.623 − 0.781i)4-s + (0.193 + 0.846i)5-s + (−0.222 + 0.974i)7-s + (−0.974 + 0.222i)8-s + (0.846 + 0.193i)10-s + (0.541 − 1.12i)11-s + (0.781 + 0.623i)14-s + (−0.222 + 0.974i)16-s + (0.541 − 0.678i)20-s + (−0.777 − 0.974i)22-s + (0.222 − 0.107i)25-s + (0.900 − 0.433i)28-s + (1.40 + 1.12i)29-s + 1.94i·31-s + (0.781 + 0.623i)32-s + ⋯ |
L(s) = 1 | + (0.433 − 0.900i)2-s + (−0.623 − 0.781i)4-s + (0.193 + 0.846i)5-s + (−0.222 + 0.974i)7-s + (−0.974 + 0.222i)8-s + (0.846 + 0.193i)10-s + (0.541 − 1.12i)11-s + (0.781 + 0.623i)14-s + (−0.222 + 0.974i)16-s + (0.541 − 0.678i)20-s + (−0.777 − 0.974i)22-s + (0.222 − 0.107i)25-s + (0.900 − 0.433i)28-s + (1.40 + 1.12i)29-s + 1.94i·31-s + (0.781 + 0.623i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.502729332\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.502729332\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.433 + 0.900i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.222 - 0.974i)T \) |
good | 5 | \( 1 + (-0.193 - 0.846i)T + (-0.900 + 0.433i)T^{2} \) |
| 11 | \( 1 + (-0.541 + 1.12i)T + (-0.623 - 0.781i)T^{2} \) |
| 13 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 17 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 29 | \( 1 + (-1.40 - 1.12i)T + (0.222 + 0.974i)T^{2} \) |
| 31 | \( 1 - 1.94iT - T^{2} \) |
| 37 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 41 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 43 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 53 | \( 1 + (-1.56 + 1.24i)T + (0.222 - 0.974i)T^{2} \) |
| 59 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 61 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.846 - 1.75i)T + (-0.623 + 0.781i)T^{2} \) |
| 79 | \( 1 - 0.445T + T^{2} \) |
| 83 | \( 1 + (-0.781 + 0.376i)T + (0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 97 | \( 1 - 1.56iT - T^{2} \) |
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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
−8.668748480650094934772638393798, −8.456463592717712572055569693144, −6.79529359804729337948407726883, −6.48604826945366230895502316808, −5.54499249989146292641485466570, −4.98076867672780404985658289451, −3.72709184483644822331654919368, −3.07888894101403832649915186550, −2.48287025638223140086078280845, −1.23990138328281876009946041289,
0.905173479951639148071386051799, 2.40622912982353829128922000849, 3.73284691530471343820472450027, 4.39250357269709809794190844371, 4.84085136285845607421408380143, 5.89852667128194701797790139800, 6.55032241723677504307022348627, 7.31307631696430860441137963471, 7.86854431581737041682165029058, 8.665299418752631944434864300281