| L(s) = 1 | + 3·3-s + 5·7-s + 9·9-s + 14·11-s + 13-s + 46·17-s + 19·19-s + 15·21-s − 46·23-s + 27·27-s + 14·29-s + 133·31-s + 42·33-s + 258·37-s + 3·39-s + 84·41-s − 167·43-s + 410·47-s − 318·49-s + 138·51-s + 456·53-s + 57·57-s − 194·59-s − 17·61-s + 45·63-s + 653·67-s − 138·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.269·7-s + 1/3·9-s + 0.383·11-s + 0.0213·13-s + 0.656·17-s + 0.229·19-s + 0.155·21-s − 0.417·23-s + 0.192·27-s + 0.0896·29-s + 0.770·31-s + 0.221·33-s + 1.14·37-s + 0.0123·39-s + 0.319·41-s − 0.592·43-s + 1.27·47-s − 0.927·49-s + 0.378·51-s + 1.18·53-s + 0.132·57-s − 0.428·59-s − 0.0356·61-s + 0.0899·63-s + 1.19·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.740482475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.740482475\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 5 T + p^{3} T^{2} \) |
| 11 | \( 1 - 14 T + p^{3} T^{2} \) |
| 13 | \( 1 - T + p^{3} T^{2} \) |
| 17 | \( 1 - 46 T + p^{3} T^{2} \) |
| 19 | \( 1 - p T + p^{3} T^{2} \) |
| 23 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 14 T + p^{3} T^{2} \) |
| 31 | \( 1 - 133 T + p^{3} T^{2} \) |
| 37 | \( 1 - 258 T + p^{3} T^{2} \) |
| 41 | \( 1 - 84 T + p^{3} T^{2} \) |
| 43 | \( 1 + 167 T + p^{3} T^{2} \) |
| 47 | \( 1 - 410 T + p^{3} T^{2} \) |
| 53 | \( 1 - 456 T + p^{3} T^{2} \) |
| 59 | \( 1 + 194 T + p^{3} T^{2} \) |
| 61 | \( 1 + 17 T + p^{3} T^{2} \) |
| 67 | \( 1 - 653 T + p^{3} T^{2} \) |
| 71 | \( 1 - 828 T + p^{3} T^{2} \) |
| 73 | \( 1 - 570 T + p^{3} T^{2} \) |
| 79 | \( 1 + 552 T + p^{3} T^{2} \) |
| 83 | \( 1 - 142 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1104 T + p^{3} T^{2} \) |
| 97 | \( 1 - 841 T + p^{3} T^{2} \) |
| 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
−10.08739957229836474837263440771, −9.432632991089721204452018249294, −8.429155463864919476718953001235, −7.75940072420533081688199362022, −6.75825932127965148456750396998, −5.69519306085952846566476914670, −4.53211551748147768776432429537, −3.52904289823064767762286064968, −2.35146608250949449895435806483, −1.01640433081996302184915470479,
1.01640433081996302184915470479, 2.35146608250949449895435806483, 3.52904289823064767762286064968, 4.53211551748147768776432429537, 5.69519306085952846566476914670, 6.75825932127965148456750396998, 7.75940072420533081688199362022, 8.429155463864919476718953001235, 9.432632991089721204452018249294, 10.08739957229836474837263440771
