L(s) = 1 | + (0.826 − 0.563i)3-s + (0.988 + 0.149i)7-s + (0.365 − 0.930i)9-s + (−0.880 − 0.702i)13-s + (−0.988 − 1.71i)19-s + (0.900 − 0.433i)21-s + (0.988 − 0.149i)25-s + (−0.222 − 0.974i)27-s + (−0.955 + 1.65i)31-s + (1.40 − 0.432i)37-s + (−1.12 − 0.0841i)39-s + (0.807 + 1.67i)43-s + (0.955 + 0.294i)49-s + (−1.78 − 0.858i)57-s + (0.574 + 1.86i)61-s + ⋯ |
L(s) = 1 | + (0.826 − 0.563i)3-s + (0.988 + 0.149i)7-s + (0.365 − 0.930i)9-s + (−0.880 − 0.702i)13-s + (−0.988 − 1.71i)19-s + (0.900 − 0.433i)21-s + (0.988 − 0.149i)25-s + (−0.222 − 0.974i)27-s + (−0.955 + 1.65i)31-s + (1.40 − 0.432i)37-s + (−1.12 − 0.0841i)39-s + (0.807 + 1.67i)43-s + (0.955 + 0.294i)49-s + (−1.78 − 0.858i)57-s + (0.574 + 1.86i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.684847208\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.684847208\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.826 + 0.563i)T \) |
| 7 | \( 1 + (-0.988 - 0.149i)T \) |
good | 5 | \( 1 + (-0.988 + 0.149i)T^{2} \) |
| 11 | \( 1 + (-0.733 + 0.680i)T^{2} \) |
| 13 | \( 1 + (0.880 + 0.702i)T + (0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (0.0747 + 0.997i)T^{2} \) |
| 19 | \( 1 + (0.988 + 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.0747 - 0.997i)T^{2} \) |
| 29 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.40 + 0.432i)T + (0.826 - 0.563i)T^{2} \) |
| 41 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.807 - 1.67i)T + (-0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 53 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 59 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 61 | \( 1 + (-0.574 - 1.86i)T + (-0.826 + 0.563i)T^{2} \) |
| 67 | \( 1 + (1.72 + 0.997i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.202 - 1.34i)T + (-0.955 + 0.294i)T^{2} \) |
| 79 | \( 1 + (0.510 - 0.294i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (-0.733 - 0.680i)T^{2} \) |
| 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
−8.837691156021263686753200172637, −8.392107805998635180250563718462, −7.45066490095718214147162013878, −7.06759891108295505005122600793, −6.01713443485117062518020988752, −4.93051234220150812504973761704, −4.33071276878621293274383970882, −2.93356985687919899681067601727, −2.40618838569707255347993940867, −1.14468434830964565516434189602,
1.75809069991171989920483295506, 2.44628118836748283791890912465, 3.78855152554227978180303856139, 4.34296198628137110931923031740, 5.13701311744572468485190985253, 6.10902698574261490604707781912, 7.32482339017168625590112001277, 7.79416292881670637269078239165, 8.543468265317399028141987791204, 9.223668148126589847823539572945