L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 4·5-s − 2·6-s − 7-s + 9-s + 8·10-s − 2·12-s + 2·13-s − 2·14-s − 4·15-s − 4·16-s − 4·17-s + 2·18-s + 3·19-s + 8·20-s + 21-s + 2·23-s + 11·25-s + 4·26-s − 27-s − 2·28-s − 6·29-s − 8·30-s − 5·31-s − 8·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 1.78·5-s − 0.816·6-s − 0.377·7-s + 1/3·9-s + 2.52·10-s − 0.577·12-s + 0.554·13-s − 0.534·14-s − 1.03·15-s − 16-s − 0.970·17-s + 0.471·18-s + 0.688·19-s + 1.78·20-s + 0.218·21-s + 0.417·23-s + 11/5·25-s + 0.784·26-s − 0.192·27-s − 0.377·28-s − 1.11·29-s − 1.46·30-s − 0.898·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.739014316\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.739014316\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−11.54728253906378153115803392514, −10.75806820233508304777352181849, −9.661688191206149098996496726670, −8.973480572648402934204682616447, −6.98676550080062361288693213459, −6.18537700457688272303978752564, −5.59539380126593624738687647200, −4.73167031165690126954598856077, −3.29412467945553736245288804676, −1.94003785572390378144466057337,
1.94003785572390378144466057337, 3.29412467945553736245288804676, 4.73167031165690126954598856077, 5.59539380126593624738687647200, 6.18537700457688272303978752564, 6.98676550080062361288693213459, 8.973480572648402934204682616447, 9.661688191206149098996496726670, 10.75806820233508304777352181849, 11.54728253906378153115803392514