| L(s) = 1 | + 3-s − 2·5-s + 9-s − 2·15-s + 2·17-s + 4·19-s − 25-s + 27-s + 6·29-s + 2·37-s − 6·41-s − 12·43-s − 2·45-s + 4·47-s − 7·49-s + 2·51-s + 6·53-s + 4·57-s + 8·59-s − 2·61-s − 4·67-s + 12·71-s + 14·73-s − 75-s + 81-s − 8·83-s − 4·85-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.516·15-s + 0.485·17-s + 0.917·19-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.328·37-s − 0.937·41-s − 1.82·43-s − 0.298·45-s + 0.583·47-s − 49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s + 1.04·59-s − 0.256·61-s − 0.488·67-s + 1.42·71-s + 1.63·73-s − 0.115·75-s + 1/9·81-s − 0.878·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.972166420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.972166420\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.277330555848996337592461167239, −7.85749416685147120585350583537, −7.11646809983531549988243962761, −6.40158815402073049865047684610, −5.30378379001641513519044692133, −4.61671937527489946798974678455, −3.62541556518995470040403604896, −3.19166320207550947955763770245, −2.02833356692568283872392736149, −0.78722055676041492708430177901,
0.78722055676041492708430177901, 2.02833356692568283872392736149, 3.19166320207550947955763770245, 3.62541556518995470040403604896, 4.61671937527489946798974678455, 5.30378379001641513519044692133, 6.40158815402073049865047684610, 7.11646809983531549988243962761, 7.85749416685147120585350583537, 8.277330555848996337592461167239
