L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 2·11-s + 13-s − 4·14-s + 16-s − 4·17-s + 2·19-s + 2·22-s + 6·23-s − 26-s + 4·28-s + 2·29-s − 4·31-s − 32-s + 4·34-s − 6·37-s − 2·38-s + 6·41-s + 8·43-s − 2·44-s − 6·46-s − 8·47-s + 9·49-s + 52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 0.603·11-s + 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.970·17-s + 0.458·19-s + 0.426·22-s + 1.25·23-s − 0.196·26-s + 0.755·28-s + 0.371·29-s − 0.718·31-s − 0.176·32-s + 0.685·34-s − 0.986·37-s − 0.324·38-s + 0.937·41-s + 1.21·43-s − 0.301·44-s − 0.884·46-s − 1.16·47-s + 9/7·49-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.676050887\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.676050887\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−8.197896443708020288283450002335, −7.46921556801921649358015610313, −6.97806275479435083663640177824, −5.99279090781092724986648644076, −5.14041814897287975498351344483, −4.67142719332311194806651455646, −3.57163965616624129570338063211, −2.51210902442494521580081324778, −1.76992862342178588205190710519, −0.78437579239845308245180225841,
0.78437579239845308245180225841, 1.76992862342178588205190710519, 2.51210902442494521580081324778, 3.57163965616624129570338063211, 4.67142719332311194806651455646, 5.14041814897287975498351344483, 5.99279090781092724986648644076, 6.97806275479435083663640177824, 7.46921556801921649358015610313, 8.197896443708020288283450002335