L(s) = 1 | + 2.76·2-s − 9.14·3-s − 0.358·4-s − 4.24·5-s − 25.2·6-s − 16.4·7-s − 23.1·8-s + 56.5·9-s − 11.7·10-s + 3.27·12-s + 13·13-s − 45.5·14-s + 38.8·15-s − 61.0·16-s − 107.·17-s + 156.·18-s + 122.·19-s + 1.52·20-s + 150.·21-s + 165.·23-s + 211.·24-s − 106.·25-s + 35.9·26-s − 270.·27-s + 5.91·28-s + 20.4·29-s + 107.·30-s + ⋯ |
L(s) = 1 | + 0.977·2-s − 1.75·3-s − 0.0448·4-s − 0.380·5-s − 1.71·6-s − 0.890·7-s − 1.02·8-s + 2.09·9-s − 0.371·10-s + 0.0788·12-s + 0.277·13-s − 0.870·14-s + 0.668·15-s − 0.953·16-s − 1.53·17-s + 2.04·18-s + 1.47·19-s + 0.0170·20-s + 1.56·21-s + 1.50·23-s + 1.79·24-s − 0.855·25-s + 0.271·26-s − 1.92·27-s + 0.0399·28-s + 0.131·29-s + 0.653·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 - 13T \) |
good | 2 | \( 1 - 2.76T + 8T^{2} \) |
| 3 | \( 1 + 9.14T + 27T^{2} \) |
| 5 | \( 1 + 4.24T + 125T^{2} \) |
| 7 | \( 1 + 16.4T + 343T^{2} \) |
| 17 | \( 1 + 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 165.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 20.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 80.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 31.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 274.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 383.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 342.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 590.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 108.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 597.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 656.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 109.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 638.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 832.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 578.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 739.T + 9.12e5T^{2} \) |
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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
−8.923139678420042065320544874776, −7.33552199746783112898038173049, −6.78926368387112632365941434238, −5.88042478090639897690344933898, −5.49892403512886879346824867720, −4.51380612535897672911505505104, −3.94795027856187663525388643481, −2.79636092664068769363912607891, −0.903319676648029382934891841531, 0,
0.903319676648029382934891841531, 2.79636092664068769363912607891, 3.94795027856187663525388643481, 4.51380612535897672911505505104, 5.49892403512886879346824867720, 5.88042478090639897690344933898, 6.78926368387112632365941434238, 7.33552199746783112898038173049, 8.923139678420042065320544874776