| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s − 4·11-s + 13-s + 2·15-s − 6·17-s + 4·19-s − 21-s − 25-s − 27-s + 6·29-s + 4·33-s − 2·35-s + 6·37-s − 39-s − 6·41-s + 4·43-s − 2·45-s + 49-s + 6·51-s − 2·53-s + 8·55-s − 4·57-s + 4·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 0.516·15-s − 1.45·17-s + 0.917·19-s − 0.218·21-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.696·33-s − 0.338·35-s + 0.986·37-s − 0.160·39-s − 0.937·41-s + 0.609·43-s − 0.298·45-s + 1/7·49-s + 0.840·51-s − 0.274·53-s + 1.07·55-s − 0.529·57-s + 0.520·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9311617331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9311617331\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.951303684850219176373370636196, −8.135980268495423139052960375931, −7.60123667345496814600041115013, −6.77795077062942325242325961481, −5.88133006849509043635940278988, −4.93803016248922274695356229711, −4.40024854312076089724865952142, −3.30940782556401294409318748139, −2.18571288805244671340846045832, −0.63309553367456222372713514327,
0.63309553367456222372713514327, 2.18571288805244671340846045832, 3.30940782556401294409318748139, 4.40024854312076089724865952142, 4.93803016248922274695356229711, 5.88133006849509043635940278988, 6.77795077062942325242325961481, 7.60123667345496814600041115013, 8.135980268495423139052960375931, 8.951303684850219176373370636196
