| L(s) = 1 | − 2-s + 4-s + 5-s + 4·7-s − 8-s − 10-s − 4·11-s − 4·14-s + 16-s − 3·17-s + 20-s + 4·22-s + 4·23-s − 4·25-s + 4·28-s + 29-s + 4·31-s − 32-s + 3·34-s + 4·35-s + 3·37-s − 40-s + 9·41-s − 8·43-s − 4·44-s − 4·46-s + 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s − 0.353·8-s − 0.316·10-s − 1.20·11-s − 1.06·14-s + 1/4·16-s − 0.727·17-s + 0.223·20-s + 0.852·22-s + 0.834·23-s − 4/5·25-s + 0.755·28-s + 0.185·29-s + 0.718·31-s − 0.176·32-s + 0.514·34-s + 0.676·35-s + 0.493·37-s − 0.158·40-s + 1.40·41-s − 1.21·43-s − 0.603·44-s − 0.589·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.607644991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.607644991\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.575876464908081598780416363800, −8.067538296094563736739558824942, −7.45847858932764171422526569300, −6.59776306979735516753353489753, −5.56558824101611044799566134166, −5.03659896196472868503573475116, −4.09198333287568191342330829239, −2.61790136137242683845415483262, −2.06097627049476068441928998048, −0.871120187799610619299598627893,
0.871120187799610619299598627893, 2.06097627049476068441928998048, 2.61790136137242683845415483262, 4.09198333287568191342330829239, 5.03659896196472868503573475116, 5.56558824101611044799566134166, 6.59776306979735516753353489753, 7.45847858932764171422526569300, 8.067538296094563736739558824942, 8.575876464908081598780416363800
