L(s) = 1 | + 2.55·2-s + 4.52·4-s − 2.44·5-s + 3.89·7-s + 6.43·8-s − 6.25·10-s − 3.82·11-s + 4.41·13-s + 9.95·14-s + 7.39·16-s + 7.77·17-s − 5.77·19-s − 11.0·20-s − 9.76·22-s + 4.71·23-s + 0.997·25-s + 11.2·26-s + 17.6·28-s + 1.05·29-s − 0.879·31-s + 6.00·32-s + 19.8·34-s − 9.54·35-s + 2.79·37-s − 14.7·38-s − 15.7·40-s − 0.154·41-s + ⋯ |
L(s) = 1 | + 1.80·2-s + 2.26·4-s − 1.09·5-s + 1.47·7-s + 2.27·8-s − 1.97·10-s − 1.15·11-s + 1.22·13-s + 2.65·14-s + 1.84·16-s + 1.88·17-s − 1.32·19-s − 2.47·20-s − 2.08·22-s + 0.983·23-s + 0.199·25-s + 2.21·26-s + 3.32·28-s + 0.194·29-s − 0.157·31-s + 1.06·32-s + 3.40·34-s − 1.61·35-s + 0.460·37-s − 2.39·38-s − 2.49·40-s − 0.0240·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.705606788\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.705606788\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 - 2.55T + 2T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 - 3.89T + 7T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 17 | \( 1 - 7.77T + 17T^{2} \) |
| 19 | \( 1 + 5.77T + 19T^{2} \) |
| 23 | \( 1 - 4.71T + 23T^{2} \) |
| 29 | \( 1 - 1.05T + 29T^{2} \) |
| 31 | \( 1 + 0.879T + 31T^{2} \) |
| 37 | \( 1 - 2.79T + 37T^{2} \) |
| 41 | \( 1 + 0.154T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 - 5.76T + 53T^{2} \) |
| 59 | \( 1 + 4.10T + 59T^{2} \) |
| 61 | \( 1 - 4.94T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 6.49T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 2.55T + 89T^{2} \) |
| 97 | \( 1 + 2.23T + 97T^{2} \) |
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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
−10.43807392157205619423842538112, −8.507028679152427391419358099351, −7.947400391496155964048428778464, −7.29419443165245705289157396095, −6.11286497296409233647778949190, −5.25855839693860939189732734282, −4.64598153347081893140279843581, −3.77979210783086930726757326662, −2.96435917465414854015847274144, −1.56283748009289920414005076144,
1.56283748009289920414005076144, 2.96435917465414854015847274144, 3.77979210783086930726757326662, 4.64598153347081893140279843581, 5.25855839693860939189732734282, 6.11286497296409233647778949190, 7.29419443165245705289157396095, 7.947400391496155964048428778464, 8.507028679152427391419358099351, 10.43807392157205619423842538112