L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 8-s + 9-s + 2·10-s + 11-s + 12-s − 2·13-s + 2·15-s + 16-s + 6·17-s + 18-s − 2·19-s + 2·20-s + 22-s + 8·23-s + 24-s − 25-s − 2·26-s + 27-s + 6·29-s + 2·30-s − 10·31-s + 32-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 + 0.301·11-s + 0.288·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.458·19-s + 0.447·20-s + 0.213·22-s + 1.66·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s + 1.11·29-s + 0.365·30-s − 1.79·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.845724701\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.845724701\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.522190661879284230008984676573, −7.48081685699030561161377096932, −7.02942490888151598539495371287, −6.10797899490534351313714676162, −5.41298048020780347133069668378, −4.76957728471121957071358966928, −3.72079657619386166260712038805, −3.00451921073101039383569404607, −2.15702452550821295309952746851, −1.21784192931379434370721694627,
1.21784192931379434370721694627, 2.15702452550821295309952746851, 3.00451921073101039383569404607, 3.72079657619386166260712038805, 4.76957728471121957071358966928, 5.41298048020780347133069668378, 6.10797899490534351313714676162, 7.02942490888151598539495371287, 7.48081685699030561161377096932, 8.522190661879284230008984676573