L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s + 6·11-s − 2·12-s + 2·13-s + 16-s − 17-s − 18-s + 4·19-s − 6·22-s + 2·24-s − 2·26-s + 4·27-s + 4·31-s − 32-s − 12·33-s + 34-s + 36-s + 4·37-s − 4·38-s − 4·39-s − 6·41-s − 8·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.577·12-s + 0.554·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 1.27·22-s + 0.408·24-s − 0.392·26-s + 0.769·27-s + 0.718·31-s − 0.176·32-s − 2.08·33-s + 0.171·34-s + 1/6·36-s + 0.657·37-s − 0.648·38-s − 0.640·39-s − 0.937·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.260412385\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.260412385\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.96179195459949, −14.21675696426482, −13.70691690525711, −13.17981817260769, −12.27846982909289, −11.91307463325978, −11.63888009746653, −11.16771703111331, −10.61158617082302, −10.00435190330315, −9.512975101667060, −8.941640436446410, −8.462880597741316, −7.835254991926610, −6.937645637791402, −6.689955952949159, −6.183087765425497, −5.617332847908276, −4.965745352632205, −4.243097566632904, −3.576629723505404, −2.883866626474754, −1.795324388763474, −1.170734226088669, −0.5741187747808603,
0.5741187747808603, 1.170734226088669, 1.795324388763474, 2.883866626474754, 3.576629723505404, 4.243097566632904, 4.965745352632205, 5.617332847908276, 6.183087765425497, 6.689955952949159, 6.937645637791402, 7.835254991926610, 8.462880597741316, 8.941640436446410, 9.512975101667060, 10.00435190330315, 10.61158617082302, 11.16771703111331, 11.63888009746653, 11.91307463325978, 12.27846982909289, 13.17981817260769, 13.70691690525711, 14.21675696426482, 14.96179195459949