L(s) = 1 | − 0.445·2-s − 0.801·4-s + 1.80·5-s − 7-s + 0.801·8-s + 9-s − 0.801·10-s − 1.24·13-s + 0.445·14-s + 0.445·16-s + 0.445·17-s − 0.445·18-s − 1.44·20-s + 1.24·23-s + 2.24·25-s + 0.554·26-s + 0.801·28-s + 0.445·31-s − 32-s − 0.198·34-s − 1.80·35-s − 0.801·36-s − 1.80·37-s + 1.44·40-s + 1.80·45-s − 0.554·46-s − 1.24·47-s + ⋯ |
L(s) = 1 | − 0.445·2-s − 0.801·4-s + 1.80·5-s − 7-s + 0.801·8-s + 9-s − 0.801·10-s − 1.24·13-s + 0.445·14-s + 0.445·16-s + 0.445·17-s − 0.445·18-s − 1.44·20-s + 1.24·23-s + 2.24·25-s + 0.554·26-s + 0.801·28-s + 0.445·31-s − 32-s − 0.198·34-s − 1.80·35-s − 0.801·36-s − 1.80·37-s + 1.44·40-s + 1.80·45-s − 0.554·46-s − 1.24·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7607023717\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7607023717\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 + 0.445T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.80T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.24T + T^{2} \) |
| 17 | \( 1 - 0.445T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.24T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 0.445T + T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.24T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.24T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 0.445T + T^{2} \) |
| 71 | \( 1 + 1.80T + T^{2} \) |
| 79 | \( 1 + 0.445T + T^{2} \) |
| 83 | \( 1 - 0.445T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−10.42071675057891263475794936832, −10.11123101763023128264037681600, −9.454364532587542877965720580982, −8.896994346669395148255496265209, −7.39437758369632863936988223320, −6.61171487719242969510236617032, −5.45557483771647096908981678655, −4.66938785864921703235026779180, −3.04027321208071451390068482415, −1.56046326235922301211901610040,
1.56046326235922301211901610040, 3.04027321208071451390068482415, 4.66938785864921703235026779180, 5.45557483771647096908981678655, 6.61171487719242969510236617032, 7.39437758369632863936988223320, 8.896994346669395148255496265209, 9.454364532587542877965720580982, 10.11123101763023128264037681600, 10.42071675057891263475794936832