L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s + 8-s + 9-s − 2·10-s + 4·11-s + 12-s − 13-s − 2·15-s + 16-s − 2·17-s + 18-s + 4·19-s − 2·20-s + 4·22-s + 4·23-s + 24-s − 25-s − 26-s + 27-s − 2·30-s − 4·31-s + 32-s + 4·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s − 0.277·13-s − 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.852·22-s + 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.365·30-s − 0.718·31-s + 0.176·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.505869841\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.505869841\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.457952627086721187707795467972, −7.50803381267876084134631679035, −7.19373302796505680349157019260, −6.31847856039725551032706751976, −5.39694937485807439770570893947, −4.43819618993787358688828552301, −3.89815222447720973515283429025, −3.20689168162354892822820190231, −2.21975332138843415159368572151, −0.989328159421161063183339069831,
0.989328159421161063183339069831, 2.21975332138843415159368572151, 3.20689168162354892822820190231, 3.89815222447720973515283429025, 4.43819618993787358688828552301, 5.39694937485807439770570893947, 6.31847856039725551032706751976, 7.19373302796505680349157019260, 7.50803381267876084134631679035, 8.457952627086721187707795467972