L(s) = 1 | + (−0.263 − 0.809i)3-s + (0.309 − 0.951i)4-s + (−0.101 − 0.0738i)5-s + (−0.393 + 1.21i)7-s + (0.222 − 0.161i)9-s + (0.728 + 0.684i)11-s − 0.851·12-s + (1.60 − 1.16i)13-s + (−0.0330 + 0.101i)15-s + (−0.809 − 0.587i)16-s + (−0.101 + 0.0738i)20-s + 1.08·21-s + (−0.304 − 0.936i)25-s + (−0.878 − 0.638i)27-s + (1.03 + 0.749i)28-s + ⋯ |
L(s) = 1 | + (−0.263 − 0.809i)3-s + (0.309 − 0.951i)4-s + (−0.101 − 0.0738i)5-s + (−0.393 + 1.21i)7-s + (0.222 − 0.161i)9-s + (0.728 + 0.684i)11-s − 0.851·12-s + (1.60 − 1.16i)13-s + (−0.0330 + 0.101i)15-s + (−0.809 − 0.587i)16-s + (−0.101 + 0.0738i)20-s + 1.08·21-s + (−0.304 − 0.936i)25-s + (−0.878 − 0.638i)27-s + (1.03 + 0.749i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.126565685\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.126565685\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.728 - 0.684i)T \) |
| 131 | \( 1 - T \) |
good | 2 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 3 | \( 1 + (0.263 + 0.809i)T + (-0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (0.101 + 0.0738i)T + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (0.393 - 1.21i)T + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-1.60 + 1.16i)T + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.541 - 1.66i)T + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + 1.85T + T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.331 + 1.01i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.101 + 0.0738i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
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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
−9.639703498693218887345302546103, −8.741219109127375659064130162835, −7.950189157895267060410262410985, −6.80261737960299837728642480968, −6.20077606041682854935358811494, −5.81193957667824407435161898903, −4.66055514538297372736396372943, −3.31952874847158390091045087929, −2.06135219026818999614812527308, −1.11117152733352728411108767987,
1.60724758205299805399170265567, 3.51123157812075346746697627819, 3.75660595533889249161798006051, 4.49517585564294615637420683599, 5.90455184232091137016912332364, 6.81476876881380230098992472922, 7.31062137107570548370146446671, 8.436961783751229518709685526659, 9.056094980959580593996994539317, 9.949283625812801487671628440389