| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 11-s − 13-s − 15-s − 17-s − 19-s − 21-s − 23-s + 25-s + 27-s + 29-s + 31-s + 33-s + 35-s + 37-s − 39-s + 41-s − 43-s − 45-s − 47-s + 49-s − 51-s + 53-s − 55-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 11-s − 13-s − 15-s − 17-s − 19-s − 21-s − 23-s + 25-s + 27-s + 29-s + 31-s + 33-s + 35-s + 37-s − 39-s + 41-s − 43-s − 45-s − 47-s + 49-s − 51-s + 53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 788 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 788 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.868676989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.868676989\) |
| \(L(1)\) |
\(\approx\) |
\(1.119146044\) |
| \(L(1)\) |
\(\approx\) |
\(1.119146044\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 197 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.0894476014984217420439569343, −21.36577471596864680649171834914, −19.975544364278993124528956941808, −19.7676589883174124808787018119, −19.31145981584764576007518213378, −18.34187985733573083855381667307, −17.1591475841618393757215998168, −16.242187106931348356242289052929, −15.53807929211245863463554456002, −14.848297287307234581203600300502, −14.085247759743112729409786616463, −13.04967175171681251873213218171, −12.38230563185916344034652161363, −11.56801429900017986761904880861, −10.30075630733872539992127289691, −9.56701257357839791399793862745, −8.69631106661472049519768437666, −7.98975658891430047920249348778, −6.91423952621109239444153073566, −6.42004804155094454423630514471, −4.53003511327044736843522909216, −4.05517126628626583807387953802, −3.0422239460969240453334267464, −2.178781419563303534749572106737, −0.615484200530402074192650187390,
0.615484200530402074192650187390, 2.178781419563303534749572106737, 3.0422239460969240453334267464, 4.05517126628626583807387953802, 4.53003511327044736843522909216, 6.42004804155094454423630514471, 6.91423952621109239444153073566, 7.98975658891430047920249348778, 8.69631106661472049519768437666, 9.56701257357839791399793862745, 10.30075630733872539992127289691, 11.56801429900017986761904880861, 12.38230563185916344034652161363, 13.04967175171681251873213218171, 14.085247759743112729409786616463, 14.848297287307234581203600300502, 15.53807929211245863463554456002, 16.242187106931348356242289052929, 17.1591475841618393757215998168, 18.34187985733573083855381667307, 19.31145981584764576007518213378, 19.7676589883174124808787018119, 19.975544364278993124528956941808, 21.36577471596864680649171834914, 22.0894476014984217420439569343
