L(s) = 1 | + 3-s − 5-s + 7-s − 2·9-s + 11-s + 2·13-s − 15-s − 6·19-s + 21-s − 23-s − 4·25-s − 5·27-s + 7·31-s + 33-s − 35-s + 5·37-s + 2·39-s + 6·41-s − 2·43-s + 2·45-s − 4·47-s + 49-s − 10·53-s − 55-s − 6·57-s − 13·59-s − 4·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.301·11-s + 0.554·13-s − 0.258·15-s − 1.37·19-s + 0.218·21-s − 0.208·23-s − 4/5·25-s − 0.962·27-s + 1.25·31-s + 0.174·33-s − 0.169·35-s + 0.821·37-s + 0.320·39-s + 0.937·41-s − 0.304·43-s + 0.298·45-s − 0.583·47-s + 1/7·49-s − 1.37·53-s − 0.134·55-s − 0.794·57-s − 1.69·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.088950340387215130696085143004, −7.40036345775193284149412579806, −6.24295764246661189206971296863, −5.99628050634450957794083313171, −4.70548843036060764874697867801, −4.15965329786557340546243869290, −3.28584981569104002278091241641, −2.47786105662843156789265544206, −1.48537664271734892133944210597, 0,
1.48537664271734892133944210597, 2.47786105662843156789265544206, 3.28584981569104002278091241641, 4.15965329786557340546243869290, 4.70548843036060764874697867801, 5.99628050634450957794083313171, 6.24295764246661189206971296863, 7.40036345775193284149412579806, 8.088950340387215130696085143004