| L(s) = 1 | + 4·7-s − 2·11-s + 13-s − 2·19-s + 2·23-s + 10·29-s + 4·31-s − 6·37-s + 6·41-s − 8·43-s − 12·47-s + 9·49-s + 14·53-s + 6·59-s − 2·61-s + 4·67-s − 14·73-s − 8·77-s − 12·83-s − 6·89-s + 4·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.603·11-s + 0.277·13-s − 0.458·19-s + 0.417·23-s + 1.85·29-s + 0.718·31-s − 0.986·37-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 9/7·49-s + 1.92·53-s + 0.781·59-s − 0.256·61-s + 0.488·67-s − 1.63·73-s − 0.911·77-s − 1.31·83-s − 0.635·89-s + 0.419·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.26325828189145, −13.08550594728734, −12.26436408700852, −11.91107548524928, −11.51828133668409, −10.98994331480085, −10.60480807042584, −10.08360353194098, −9.776162744954829, −8.783844466412643, −8.440683158094603, −8.358926614890310, −7.662418296292362, −7.130529503764530, −6.647266936441539, −6.070120530241999, −5.361095648826409, −5.055718960020525, −4.516058095818820, −4.121450768650931, −3.285671711805120, −2.702158817382012, −2.166905176189787, −1.436061680973018, −0.9958293788857313, 0,
0.9958293788857313, 1.436061680973018, 2.166905176189787, 2.702158817382012, 3.285671711805120, 4.121450768650931, 4.516058095818820, 5.055718960020525, 5.361095648826409, 6.070120530241999, 6.647266936441539, 7.130529503764530, 7.662418296292362, 8.358926614890310, 8.440683158094603, 8.783844466412643, 9.776162744954829, 10.08360353194098, 10.60480807042584, 10.98994331480085, 11.51828133668409, 11.91107548524928, 12.26436408700852, 13.08550594728734, 13.26325828189145
