| L(s) = 1 | + 4.53·2-s + 12.5·4-s − 7·7-s + 20.7·8-s + 54.0·11-s − 75.2·13-s − 31.7·14-s − 6.60·16-s − 71.2·17-s − 65.5·19-s + 245.·22-s + 125.·23-s − 341.·26-s − 87.9·28-s − 190.·29-s − 193.·31-s − 195.·32-s − 323.·34-s − 114.·37-s − 297.·38-s − 216.·41-s + 413.·43-s + 679.·44-s + 570.·46-s − 113.·47-s + 49·49-s − 945.·52-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 1.57·4-s − 0.377·7-s + 0.915·8-s + 1.48·11-s − 1.60·13-s − 0.606·14-s − 0.103·16-s − 1.01·17-s − 0.791·19-s + 2.37·22-s + 1.13·23-s − 2.57·26-s − 0.593·28-s − 1.21·29-s − 1.11·31-s − 1.08·32-s − 1.62·34-s − 0.509·37-s − 1.26·38-s − 0.826·41-s + 1.46·43-s + 2.32·44-s + 1.82·46-s − 0.352·47-s + 0.142·49-s − 2.52·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 4.53T + 8T^{2} \) |
| 11 | \( 1 - 54.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 75.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 71.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 65.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 190.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 193.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 216.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 113.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 584.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 203.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 162.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 477.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 822.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 798.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 468.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 310.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.31e3T + 9.12e5T^{2} \) |
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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
−8.941664626530402052209743965201, −7.27608393770094668537368206929, −6.92500044824474366830360661971, −6.08445837208276198508666334141, −5.21781657259045096229567898352, −4.39137653105078944536123181915, −3.77033592552118968377450768158, −2.73288405632729858763018071585, −1.83034871472492507952397496032, 0,
1.83034871472492507952397496032, 2.73288405632729858763018071585, 3.77033592552118968377450768158, 4.39137653105078944536123181915, 5.21781657259045096229567898352, 6.08445837208276198508666334141, 6.92500044824474366830360661971, 7.27608393770094668537368206929, 8.941664626530402052209743965201
