L(s) = 1 | + (0.733 + 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.0747 + 0.997i)7-s + (−0.623 + 0.781i)8-s + (0.955 + 0.294i)9-s + (1.88 − 0.582i)11-s + (−0.222 + 0.974i)13-s + (−0.733 + 0.680i)14-s + (−0.988 + 0.149i)16-s + (−1.21 − 0.825i)17-s + (0.5 + 0.866i)18-s + (0.955 − 1.65i)19-s + (1.78 + 0.858i)22-s + (−0.733 + 0.680i)25-s + (−0.826 + 0.563i)26-s + ⋯ |
L(s) = 1 | + (0.733 + 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.0747 + 0.997i)7-s + (−0.623 + 0.781i)8-s + (0.955 + 0.294i)9-s + (1.88 − 0.582i)11-s + (−0.222 + 0.974i)13-s + (−0.733 + 0.680i)14-s + (−0.988 + 0.149i)16-s + (−1.21 − 0.825i)17-s + (0.5 + 0.866i)18-s + (0.955 − 1.65i)19-s + (1.78 + 0.858i)22-s + (−0.733 + 0.680i)25-s + (−0.826 + 0.563i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.971269156\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.971269156\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.733 - 0.680i)T \) |
| 7 | \( 1 + (0.0747 - 0.997i)T \) |
| 13 | \( 1 + (0.222 - 0.974i)T \) |
good | 3 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 5 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 11 | \( 1 + (-1.88 + 0.582i)T + (0.826 - 0.563i)T^{2} \) |
| 17 | \( 1 + (1.21 + 0.825i)T + (0.365 + 0.930i)T^{2} \) |
| 19 | \( 1 + (-0.955 + 1.65i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 29 | \( 1 + (0.658 - 0.317i)T + (0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 41 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.535 - 0.496i)T + (0.0747 + 0.997i)T^{2} \) |
| 53 | \( 1 + (0.0747 + 0.997i)T + (-0.988 + 0.149i)T^{2} \) |
| 59 | \( 1 + (0.0546 - 0.139i)T + (-0.733 - 0.680i)T^{2} \) |
| 61 | \( 1 + (0.0747 - 0.997i)T + (-0.988 - 0.149i)T^{2} \) |
| 67 | \( 1 + (-0.826 - 1.43i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (1.32 + 0.636i)T + (0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.162648339174327843680451762954, −8.716848271084967264650322612726, −7.35788429416748640518833750184, −6.96861491216755622212274079118, −6.30653907069600823464910287532, −5.39168660498905236130696481738, −4.55703298711824441359274292092, −3.93926009827848921917412651368, −2.83857748230021313836597191819, −1.78521352804658724478204263335,
1.19014187577516976175355778211, 1.90342440950362705360193897599, 3.55372369894177776138416550809, 3.94240525984601932803242490623, 4.54098808356664476902084794213, 5.77102195076990745242029209656, 6.50730338370372884608169514792, 7.11060671349670698156256756420, 8.012815685301672768150031809535, 9.262267248932651395768080102754