| L(s) = 1 | + 3-s − 3.24·7-s + 9-s − 0.460·11-s + 13-s − 4.26·17-s + 1.70·19-s − 3.24·21-s + 8.38·23-s + 27-s + 3.34·29-s − 7.87·31-s − 0.460·33-s + 7.46·37-s + 39-s − 8.78·41-s − 3.89·43-s + 0.751·47-s + 3.55·49-s − 4.26·51-s − 8.70·53-s + 1.70·57-s − 3.27·59-s − 8.07·61-s − 3.24·63-s + 4.95·67-s + 8.38·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.22·7-s + 0.333·9-s − 0.138·11-s + 0.277·13-s − 1.03·17-s + 0.392·19-s − 0.708·21-s + 1.74·23-s + 0.192·27-s + 0.620·29-s − 1.41·31-s − 0.0802·33-s + 1.22·37-s + 0.160·39-s − 1.37·41-s − 0.593·43-s + 0.109·47-s + 0.507·49-s − 0.596·51-s − 1.19·53-s + 0.226·57-s − 0.426·59-s − 1.03·61-s − 0.409·63-s + 0.605·67-s + 1.00·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3.24T + 7T^{2} \) |
| 11 | \( 1 + 0.460T + 11T^{2} \) |
| 17 | \( 1 + 4.26T + 17T^{2} \) |
| 19 | \( 1 - 1.70T + 19T^{2} \) |
| 23 | \( 1 - 8.38T + 23T^{2} \) |
| 29 | \( 1 - 3.34T + 29T^{2} \) |
| 31 | \( 1 + 7.87T + 31T^{2} \) |
| 37 | \( 1 - 7.46T + 37T^{2} \) |
| 41 | \( 1 + 8.78T + 41T^{2} \) |
| 43 | \( 1 + 3.89T + 43T^{2} \) |
| 47 | \( 1 - 0.751T + 47T^{2} \) |
| 53 | \( 1 + 8.70T + 53T^{2} \) |
| 59 | \( 1 + 3.27T + 59T^{2} \) |
| 61 | \( 1 + 8.07T + 61T^{2} \) |
| 67 | \( 1 - 4.95T + 67T^{2} \) |
| 71 | \( 1 - 8.23T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 - 7.40T + 83T^{2} \) |
| 89 | \( 1 + 9.44T + 89T^{2} \) |
| 97 | \( 1 + 2.49T + 97T^{2} \) |
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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
−7.46752314459156243723744748820, −6.71454363747251563286837585232, −6.42466918967995083272836673623, −5.35189552246265165594389930967, −4.66445312053684309837626136192, −3.68696826474404006952361730615, −3.14250107078741104455590763966, −2.44015428937321343220332589783, −1.29964482889531259012845619998, 0,
1.29964482889531259012845619998, 2.44015428937321343220332589783, 3.14250107078741104455590763966, 3.68696826474404006952361730615, 4.66445312053684309837626136192, 5.35189552246265165594389930967, 6.42466918967995083272836673623, 6.71454363747251563286837585232, 7.46752314459156243723744748820
