| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 2·13-s + 15-s − 3·17-s − 4·19-s − 21-s − 2·23-s − 4·25-s − 27-s − 8·29-s − 10·31-s − 35-s − 6·37-s + 2·39-s + 6·41-s + 7·43-s − 45-s + 13·47-s + 49-s + 3·51-s − 8·53-s + 4·57-s − 11·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 0.727·17-s − 0.917·19-s − 0.218·21-s − 0.417·23-s − 4/5·25-s − 0.192·27-s − 1.48·29-s − 1.79·31-s − 0.169·35-s − 0.986·37-s + 0.320·39-s + 0.937·41-s + 1.06·43-s − 0.149·45-s + 1.89·47-s + 1/7·49-s + 0.420·51-s − 1.09·53-s + 0.529·57-s − 1.43·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7291149927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7291149927\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−16.80816680768330, −15.88740940601369, −15.59586795644051, −15.01215215371740, −14.29573414897331, −13.88306120244499, −12.83450100912665, −12.66060981134421, −12.00144434159200, −11.13372297255386, −11.03869282755785, −10.36194223239378, −9.344026324619835, −9.137741749383064, −8.120276753163335, −7.568780697645612, −7.068688241572290, −6.222596275152911, −5.591116099198571, −4.955284125143352, −4.064032003755149, −3.754959531157618, −2.352088234409118, −1.807895813620983, −0.3987946826312976,
0.3987946826312976, 1.807895813620983, 2.352088234409118, 3.754959531157618, 4.064032003755149, 4.955284125143352, 5.591116099198571, 6.222596275152911, 7.068688241572290, 7.568780697645612, 8.120276753163335, 9.137741749383064, 9.344026324619835, 10.36194223239378, 11.03869282755785, 11.13372297255386, 12.00144434159200, 12.66060981134421, 12.83450100912665, 13.88306120244499, 14.29573414897331, 15.01215215371740, 15.59586795644051, 15.88740940601369, 16.80816680768330
