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
\(L(1)\) |
\(\approx\) |
\(2.248658230\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.248658230\) |
\(L(\frac{3}{2})\) |
|
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
−7.87498419555034890742165610765, −7.29485373851164431988882778158, −6.60290116620330028250116272132, −5.77144927324143348913963681994, −5.48331297526468208290874126097, −4.32578104712105166619371068892, −3.62643806926958717220404425149, −2.94308915875784264076705004238, −2.18984263408584192767406707815, −0.68431411467238404496665432852,
0.68431411467238404496665432852, 2.18984263408584192767406707815, 2.94308915875784264076705004238, 3.62643806926958717220404425149, 4.32578104712105166619371068892, 5.48331297526468208290874126097, 5.77144927324143348913963681994, 6.60290116620330028250116272132, 7.29485373851164431988882778158, 7.87498419555034890742165610765