L(s) = 1 | + (0.809 + 0.587i)5-s + (0.309 − 0.951i)9-s + (0.5 − 1.53i)13-s + (−0.5 − 0.363i)17-s + (0.309 + 0.951i)25-s + (0.5 − 0.363i)29-s + (−0.190 + 0.587i)37-s + (−0.5 + 1.53i)41-s + (0.809 − 0.587i)45-s + 49-s + (−1.30 + 0.951i)53-s + (0.5 + 1.53i)61-s + (1.30 − 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 + (0.5 − 1.53i)13-s + (−0.5 − 0.363i)17-s + (0.309 + 0.951i)25-s + (0.5 − 0.363i)29-s + (−0.190 + 0.587i)37-s + (−0.5 + 1.53i)41-s + (0.809 − 0.587i)45-s + 49-s + (−1.30 + 0.951i)53-s + (0.5 + 1.53i)61-s + (1.30 − 0.951i)65-s + (−0.5 − 1.53i)73-s + (−0.809 − 0.587i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.317301027\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.317301027\) |
\(L(1)\) |
|
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 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 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 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 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 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 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
−9.666962531555811282293756592510, −8.876469138694748065824052559627, −7.983798737799424788397465914883, −7.04094524146893325818338596307, −6.28398985426557245456819981092, −5.69520076277469174969051435177, −4.60621590431531897335432868065, −3.38310452449046606068916386727, −2.70502670592038799258822071380, −1.22470948941782794062876310414,
1.60337412170165041262684511696, 2.28758774429350980591967667312, 3.88433363542550711663554815012, 4.70455789732372464393497721490, 5.46764742139204314842131343284, 6.44551046213581650169088745613, 7.10202434112537957882218144565, 8.257171250378293586665876277720, 8.855415774501323226738822145814, 9.544384037904945321267680352639