| L(s) = 1 | + 3-s − 3·7-s + 9-s + 11-s − 13-s + 5·17-s + 8·19-s − 3·21-s + 27-s + 29-s − 3·31-s + 33-s + 8·37-s − 39-s − 2·41-s + 8·43-s − 11·47-s + 2·49-s + 5·51-s + 11·53-s + 8·57-s − 5·59-s + 61-s − 3·63-s + 3·67-s − 16·71-s + 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 1.21·17-s + 1.83·19-s − 0.654·21-s + 0.192·27-s + 0.185·29-s − 0.538·31-s + 0.174·33-s + 1.31·37-s − 0.160·39-s − 0.312·41-s + 1.21·43-s − 1.60·47-s + 2/7·49-s + 0.700·51-s + 1.51·53-s + 1.05·57-s − 0.650·59-s + 0.128·61-s − 0.377·63-s + 0.366·67-s − 1.89·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.606989337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.606989337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.98250115516181, −15.58377821516295, −14.65474254374882, −14.47360770950634, −13.80194373537948, −13.16167714556641, −12.83092242907484, −11.99697384874591, −11.76481661192058, −10.86415340395933, −10.04080812515258, −9.715689267682443, −9.339685379690898, −8.604834120057214, −7.797117597370585, −7.380252383452715, −6.792355923043476, −5.939389651641131, −5.497321471833305, −4.586010560648692, −3.758820452730212, −3.151635777719381, −2.748239032628001, −1.556757835893199, −0.7054304255010389,
0.7054304255010389, 1.556757835893199, 2.748239032628001, 3.151635777719381, 3.758820452730212, 4.586010560648692, 5.497321471833305, 5.939389651641131, 6.792355923043476, 7.380252383452715, 7.797117597370585, 8.604834120057214, 9.339685379690898, 9.715689267682443, 10.04080812515258, 10.86415340395933, 11.76481661192058, 11.99697384874591, 12.83092242907484, 13.16167714556641, 13.80194373537948, 14.47360770950634, 14.65474254374882, 15.58377821516295, 15.98250115516181
