| L(s) = 1 | − 2·3-s + 16·5-s − 23·9-s − 24·11-s + 68·13-s − 32·15-s − 54·17-s − 46·19-s − 176·23-s + 131·25-s + 100·27-s − 174·29-s − 116·31-s + 48·33-s + 74·37-s − 136·39-s + 10·41-s + 480·43-s − 368·45-s − 572·47-s + 108·51-s − 162·53-s − 384·55-s + 92·57-s − 86·59-s + 904·61-s + 1.08e3·65-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 1.43·5-s − 0.851·9-s − 0.657·11-s + 1.45·13-s − 0.550·15-s − 0.770·17-s − 0.555·19-s − 1.59·23-s + 1.04·25-s + 0.712·27-s − 1.11·29-s − 0.672·31-s + 0.253·33-s + 0.328·37-s − 0.558·39-s + 0.0380·41-s + 1.70·43-s − 1.21·45-s − 1.77·47-s + 0.296·51-s − 0.419·53-s − 0.941·55-s + 0.213·57-s − 0.189·59-s + 1.89·61-s + 2.07·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 68 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 46 T + p^{3} T^{2} \) |
| 23 | \( 1 + 176 T + p^{3} T^{2} \) |
| 29 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 116 T + p^{3} T^{2} \) |
| 37 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 41 | \( 1 - 10 T + p^{3} T^{2} \) |
| 43 | \( 1 - 480 T + p^{3} T^{2} \) |
| 47 | \( 1 + 572 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 86 T + p^{3} T^{2} \) |
| 61 | \( 1 - 904 T + p^{3} T^{2} \) |
| 67 | \( 1 + 660 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1024 T + p^{3} T^{2} \) |
| 73 | \( 1 + 770 T + p^{3} T^{2} \) |
| 79 | \( 1 - 904 T + p^{3} T^{2} \) |
| 83 | \( 1 - 682 T + p^{3} T^{2} \) |
| 89 | \( 1 - 102 T + p^{3} T^{2} \) |
| 97 | \( 1 - 218 T + p^{3} T^{2} \) |
| 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
−9.446688206050273763716408443496, −8.761055208771752190471987504805, −7.87362287080087552373636451101, −6.42026568292406103583397386440, −5.97961176961872151545692822962, −5.30444415160046039575712572772, −3.95860606844688018440414011806, −2.58113382635076576522156403954, −1.65110501896567048049317094266, 0,
1.65110501896567048049317094266, 2.58113382635076576522156403954, 3.95860606844688018440414011806, 5.30444415160046039575712572772, 5.97961176961872151545692822962, 6.42026568292406103583397386440, 7.87362287080087552373636451101, 8.761055208771752190471987504805, 9.446688206050273763716408443496
