| L(s) = 1 | + 7-s − 3·9-s + 4·11-s − 4·13-s + 8·17-s + 2·19-s + 23-s + 2·29-s + 6·31-s + 10·37-s + 6·41-s − 8·43-s + 6·47-s + 49-s − 2·53-s + 10·61-s − 3·63-s + 8·67-s + 12·71-s − 6·73-s + 4·77-s + 9·81-s + 2·83-s + 12·89-s − 4·91-s − 12·97-s − 12·99-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s + 1.20·11-s − 1.10·13-s + 1.94·17-s + 0.458·19-s + 0.208·23-s + 0.371·29-s + 1.07·31-s + 1.64·37-s + 0.937·41-s − 1.21·43-s + 0.875·47-s + 1/7·49-s − 0.274·53-s + 1.28·61-s − 0.377·63-s + 0.977·67-s + 1.42·71-s − 0.702·73-s + 0.455·77-s + 81-s + 0.219·83-s + 1.27·89-s − 0.419·91-s − 1.21·97-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.261599570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.261599570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.27141198518517, −14.01044366566373, −13.27596289351987, −12.54887060005992, −12.10051075781159, −11.76613321495227, −11.38682189467966, −10.73709674829893, −10.00175400726736, −9.639935588719357, −9.269728360735353, −8.441304168916575, −8.079953089144072, −7.577116535230983, −6.959254815177338, −6.329114952779350, −5.776587279431511, −5.253280613047466, −4.721453600484380, −4.001921072600272, −3.349177508438600, −2.779663494198660, −2.162817824831169, −1.117052627851532, −0.7217867600571541,
0.7217867600571541, 1.117052627851532, 2.162817824831169, 2.779663494198660, 3.349177508438600, 4.001921072600272, 4.721453600484380, 5.253280613047466, 5.776587279431511, 6.329114952779350, 6.959254815177338, 7.577116535230983, 8.079953089144072, 8.441304168916575, 9.269728360735353, 9.639935588719357, 10.00175400726736, 10.73709674829893, 11.38682189467966, 11.76613321495227, 12.10051075781159, 12.54887060005992, 13.27596289351987, 14.01044366566373, 14.27141198518517
