L(s) = 1 | + (−0.781 − 0.623i)3-s + (−0.997 + 0.0747i)4-s + (−0.733 − 0.680i)7-s + (0.222 + 0.974i)9-s + (0.826 + 0.563i)12-s + (−0.930 − 0.365i)13-s + (0.988 − 0.149i)16-s + (−1.25 + 1.25i)19-s + (0.149 + 0.988i)21-s + (0.294 − 0.955i)25-s + (0.433 − 0.900i)27-s + (0.781 + 0.623i)28-s + (0.488 + 1.82i)31-s + (−0.294 − 0.955i)36-s + (0.988 + 1.14i)37-s + ⋯ |
L(s) = 1 | + (−0.781 − 0.623i)3-s + (−0.997 + 0.0747i)4-s + (−0.733 − 0.680i)7-s + (0.222 + 0.974i)9-s + (0.826 + 0.563i)12-s + (−0.930 − 0.365i)13-s + (0.988 − 0.149i)16-s + (−1.25 + 1.25i)19-s + (0.149 + 0.988i)21-s + (0.294 − 0.955i)25-s + (0.433 − 0.900i)27-s + (0.781 + 0.623i)28-s + (0.488 + 1.82i)31-s + (−0.294 − 0.955i)36-s + (0.988 + 1.14i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3445947739\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3445947739\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.781 + 0.623i)T \) |
| 7 | \( 1 + (0.733 + 0.680i)T \) |
| 13 | \( 1 + (0.930 + 0.365i)T \) |
good | 2 | \( 1 + (0.997 - 0.0747i)T^{2} \) |
| 5 | \( 1 + (-0.294 + 0.955i)T^{2} \) |
| 11 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 17 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 19 | \( 1 + (1.25 - 1.25i)T - iT^{2} \) |
| 23 | \( 1 + (0.365 - 0.930i)T^{2} \) |
| 29 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 31 | \( 1 + (-0.488 - 1.82i)T + (-0.866 + 0.5i)T^{2} \) |
| 37 | \( 1 + (-0.988 - 1.14i)T + (-0.149 + 0.988i)T^{2} \) |
| 41 | \( 1 + (-0.294 + 0.955i)T^{2} \) |
| 43 | \( 1 + (0.0878 + 0.582i)T + (-0.955 + 0.294i)T^{2} \) |
| 47 | \( 1 + (-0.563 - 0.826i)T^{2} \) |
| 53 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 59 | \( 1 + (-0.680 + 0.733i)T^{2} \) |
| 61 | \( 1 + (-0.716 - 1.48i)T + (-0.623 + 0.781i)T^{2} \) |
| 67 | \( 1 + (-0.158 - 0.158i)T + iT^{2} \) |
| 71 | \( 1 + (-0.930 - 0.365i)T^{2} \) |
| 73 | \( 1 + (-0.918 - 1.73i)T + (-0.563 + 0.826i)T^{2} \) |
| 79 | \( 1 + (-0.433 - 0.751i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 89 | \( 1 + (0.563 - 0.826i)T^{2} \) |
| 97 | \( 1 + (-0.241 - 0.902i)T + (-0.866 + 0.5i)T^{2} \) |
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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
−9.815761948500674590086564221853, −8.519185432920094627286678393788, −8.046807884146526696383841369122, −7.05380725034434827771923806810, −6.41580932489607521882296467372, −5.51957229059931561967749368683, −4.67410112192391362876636928954, −3.93685684583440056276490995186, −2.67093078501199033447484940790, −1.08858746348612697965074282377,
0.34187011361883567625140769499, 2.42649828980819399555192565490, 3.64778863640515897870121289467, 4.50799709842235773335157039955, 5.10464435011624995194290250109, 5.99220191900392174376895920539, 6.64001049082907985563264952383, 7.74847667443580863481871454703, 8.869891164516890817038326491951, 9.414545088874169132519432081468