L(s) = 1 | + (−1.79 − 0.881i)2-s + (1.82 + 2.36i)4-s + (−0.998 − 0.0570i)7-s + (−0.787 − 3.88i)8-s + (0.309 − 0.951i)9-s + (−0.985 + 0.170i)11-s + (1.73 + 0.981i)14-s + (−1.25 + 4.77i)16-s + (−1.39 + 1.43i)18-s + (1.91 + 0.562i)22-s + (−0.610 − 0.392i)23-s + (0.0855 − 0.996i)25-s + (−1.68 − 2.46i)28-s + (−0.126 − 1.46i)29-s + (3.85 − 4.44i)32-s + ⋯ |
L(s) = 1 | + (−1.79 − 0.881i)2-s + (1.82 + 2.36i)4-s + (−0.998 − 0.0570i)7-s + (−0.787 − 3.88i)8-s + (0.309 − 0.951i)9-s + (−0.985 + 0.170i)11-s + (1.73 + 0.981i)14-s + (−1.25 + 4.77i)16-s + (−1.39 + 1.43i)18-s + (1.91 + 0.562i)22-s + (−0.610 − 0.392i)23-s + (0.0855 − 0.996i)25-s + (−1.68 − 2.46i)28-s + (−0.126 − 1.46i)29-s + (3.85 − 4.44i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2297993713\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2297993713\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.998 + 0.0570i)T \) |
| 11 | \( 1 + (0.985 - 0.170i)T \) |
good | 2 | \( 1 + (1.79 + 0.881i)T + (0.610 + 0.791i)T^{2} \) |
| 3 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (-0.0855 + 0.996i)T^{2} \) |
| 13 | \( 1 + (0.362 + 0.931i)T^{2} \) |
| 17 | \( 1 + (-0.696 - 0.717i)T^{2} \) |
| 19 | \( 1 + (-0.897 - 0.441i)T^{2} \) |
| 23 | \( 1 + (0.610 + 0.392i)T + (0.415 + 0.909i)T^{2} \) |
| 29 | \( 1 + (0.126 + 1.46i)T + (-0.985 + 0.170i)T^{2} \) |
| 31 | \( 1 + (0.998 + 0.0570i)T^{2} \) |
| 37 | \( 1 + (1.52 + 0.543i)T + (0.774 + 0.633i)T^{2} \) |
| 41 | \( 1 + (0.564 - 0.825i)T^{2} \) |
| 43 | \( 1 + (0.211 - 0.462i)T + (-0.654 - 0.755i)T^{2} \) |
| 47 | \( 1 + (0.0285 - 0.999i)T^{2} \) |
| 53 | \( 1 + (0.262 + 0.998i)T + (-0.870 + 0.491i)T^{2} \) |
| 59 | \( 1 + (0.564 + 0.825i)T^{2} \) |
| 61 | \( 1 + (-0.610 + 0.791i)T^{2} \) |
| 67 | \( 1 + (-0.262 + 1.82i)T + (-0.959 - 0.281i)T^{2} \) |
| 71 | \( 1 + (0.582 - 0.966i)T + (-0.466 - 0.884i)T^{2} \) |
| 73 | \( 1 + (0.736 - 0.676i)T^{2} \) |
| 79 | \( 1 + (0.0567 - 0.00651i)T + (0.974 - 0.226i)T^{2} \) |
| 83 | \( 1 + (-0.198 - 0.980i)T^{2} \) |
| 89 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 97 | \( 1 + (-0.0855 - 0.996i)T^{2} \) |
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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
−10.02055235659716354930615761703, −9.397568106211714467887548631599, −8.536449924647267060772385033004, −7.76148952481253004729556300036, −6.85534292952086983297780389541, −6.17949757683609010110275517103, −4.03983055127589528906295934673, −3.12311175001856071969195867352, −2.13485817592103825058347978422, −0.38984403252120521967286600216,
1.69517611571343726449504776786, 2.94025981514768206751402305727, 5.15548144336446820864134108421, 5.76410885544614402680550607639, 6.91690337919034182262994258349, 7.38926816526430123917682932995, 8.276486047253579668232219623356, 8.996265798190268132278018639137, 9.848784009565105577864090868487, 10.46888153791329000754902394405