| L(s) = 1 | − 2-s + 4-s − 3·5-s − 7-s − 8-s + 3·10-s − 11-s − 13-s + 14-s + 16-s − 7·17-s + 19-s − 3·20-s + 22-s + 7·23-s + 4·25-s + 26-s − 28-s − 3·29-s − 32-s + 7·34-s + 3·35-s − 5·37-s − 38-s + 3·40-s − 4·41-s + 11·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.377·7-s − 0.353·8-s + 0.948·10-s − 0.301·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.69·17-s + 0.229·19-s − 0.670·20-s + 0.213·22-s + 1.45·23-s + 4/5·25-s + 0.196·26-s − 0.188·28-s − 0.557·29-s − 0.176·32-s + 1.20·34-s + 0.507·35-s − 0.821·37-s − 0.162·38-s + 0.474·40-s − 0.624·41-s + 1.67·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5875381626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5875381626\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−9.042633658738900996408374505805, −8.794485082046221789728545251895, −7.71886793500763207100130548275, −7.20481276136351597364288703843, −6.49587991264233814474632939114, −5.23206193743426850928425903169, −4.25945441857471440765324112942, −3.33711352613587716311643994647, −2.28508171651994554248500999764, −0.56722170613835914134961311701,
0.56722170613835914134961311701, 2.28508171651994554248500999764, 3.33711352613587716311643994647, 4.25945441857471440765324112942, 5.23206193743426850928425903169, 6.49587991264233814474632939114, 7.20481276136351597364288703843, 7.71886793500763207100130548275, 8.794485082046221789728545251895, 9.042633658738900996408374505805
