| L(s) = 1 | + 2-s + 4-s + 8-s − 11-s + 1.41·13-s + 16-s − 4·17-s + 5.41·19-s − 22-s − 7.65·23-s − 5·25-s + 1.41·26-s + 5.65·29-s − 3.07·31-s + 32-s − 4·34-s + 3.65·37-s + 5.41·38-s − 9.65·41-s − 10·43-s − 44-s − 7.65·46-s + 1.41·47-s − 5·50-s + 1.41·52-s + 3.65·53-s + 5.65·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.353·8-s − 0.301·11-s + 0.392·13-s + 0.250·16-s − 0.970·17-s + 1.24·19-s − 0.213·22-s − 1.59·23-s − 25-s + 0.277·26-s + 1.05·29-s − 0.551·31-s + 0.176·32-s − 0.685·34-s + 0.601·37-s + 0.878·38-s − 1.50·41-s − 1.52·43-s − 0.150·44-s − 1.12·46-s + 0.206·47-s − 0.707·50-s + 0.196·52-s + 0.502·53-s + 0.742·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 5.41T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 + 3.07T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 + 9.65T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 - 3.65T + 53T^{2} \) |
| 59 | \( 1 + 6.82T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 2.58T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
| 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
−7.31821021334566361084735667859, −6.44713373955690254987157833210, −6.02133623511271840062019651302, −5.21806493573890461280459696776, −4.59075600414365518391393782159, −3.81224155942979933118912245834, −3.16970989580926244640150791255, −2.24564842612789629388186244474, −1.46740652730559253332895514935, 0,
1.46740652730559253332895514935, 2.24564842612789629388186244474, 3.16970989580926244640150791255, 3.81224155942979933118912245834, 4.59075600414365518391393782159, 5.21806493573890461280459696776, 6.02133623511271840062019651302, 6.44713373955690254987157833210, 7.31821021334566361084735667859
