L(s) = 1 | + (0.809 + 0.587i)5-s + (−0.309 + 0.951i)9-s + (1.11 + 0.363i)13-s + (0.5 + 0.363i)17-s + (0.309 + 0.951i)25-s + (−1.11 − 1.53i)29-s + (−1.80 − 0.587i)37-s + (−0.5 + 1.53i)41-s + (−0.809 + 0.587i)45-s + 49-s + (0.690 + 0.951i)53-s + (1.11 − 0.363i)61-s + (0.690 + 0.951i)65-s + (0.5 + 1.53i)73-s + (−0.809 − 0.587i)81-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)5-s + (−0.309 + 0.951i)9-s + (1.11 + 0.363i)13-s + (0.5 + 0.363i)17-s + (0.309 + 0.951i)25-s + (−1.11 − 1.53i)29-s + (−1.80 − 0.587i)37-s + (−0.5 + 1.53i)41-s + (−0.809 + 0.587i)45-s + 49-s + (0.690 + 0.951i)53-s + (1.11 − 0.363i)61-s + (0.690 + 0.951i)65-s + (0.5 + 1.53i)73-s + (−0.809 − 0.587i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.464733528\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.464733528\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.809 - 0.587i)T \) |
good | 3 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
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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
−8.900710867470391428199904589881, −8.277755405831053277744817070755, −7.44856314488837726622124437013, −6.68578085724934243233203831304, −5.80789596988122099964611496146, −5.45873455142173248204850709313, −4.24958058728688162805137260719, −3.37337639601317900676233575704, −2.35589691724804671962656676417, −1.56617847762841779861088936776,
0.954130413471926463443190414300, 1.97463176479611492721024713556, 3.28971244864316759972261879328, 3.84092321382740838585372965437, 5.26522115555883223079368709631, 5.49061368615348345455993588123, 6.46666624929956502456626490401, 7.07235119353696800808719586858, 8.195883463696030309640478087075, 8.967102935273781400584814350917