| L(s) = 1 | + 2i·3-s + 6i·7-s + 23·9-s + 60·11-s − 50i·13-s − 30i·17-s − 40·19-s − 12·21-s − 178i·23-s + 100i·27-s − 166·29-s + 20·31-s + 120i·33-s + 10i·37-s + 100·39-s + ⋯ |
| L(s) = 1 | + 0.384i·3-s + 0.323i·7-s + 0.851·9-s + 1.64·11-s − 1.06i·13-s − 0.428i·17-s − 0.482·19-s − 0.124·21-s − 1.61i·23-s + 0.712i·27-s − 1.06·29-s + 0.115·31-s + 0.633i·33-s + 0.0444i·37-s + 0.410·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.315827793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.315827793\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2iT - 27T^{2} \) |
| 7 | \( 1 - 6iT - 343T^{2} \) |
| 11 | \( 1 - 60T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 30iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 40T + 6.85e3T^{2} \) |
| 23 | \( 1 + 178iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 166T + 2.43e4T^{2} \) |
| 31 | \( 1 - 20T + 2.97e4T^{2} \) |
| 37 | \( 1 - 10iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 250T + 6.89e4T^{2} \) |
| 43 | \( 1 + 142iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 214iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 490iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 800T + 2.05e5T^{2} \) |
| 61 | \( 1 - 250T + 2.26e5T^{2} \) |
| 67 | \( 1 + 774iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 100T + 3.57e5T^{2} \) |
| 73 | \( 1 - 230iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.32e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 982iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 874T + 7.04e5T^{2} \) |
| 97 | \( 1 + 310iT - 9.12e5T^{2} \) |
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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
−9.816259057052212511119592589013, −9.032732472602482014352527687217, −8.263720276516943120000263714428, −7.08377117381034597765050253729, −6.40992542856802199887632623207, −5.28343637804395783537976304326, −4.29122235409380352279644887160, −3.47523689191275757292368106728, −2.05568073741559131502295398844, −0.70975310124047142595551272372,
1.18402841078320556941008993391, 1.92230138412185716536512001364, 3.78337878194168354118339153295, 4.19866277594699220204188359687, 5.63468931620937173438701790792, 6.78135814630263318874160444823, 7.04564683381188809825546020486, 8.213910924648798787863582097056, 9.254236435359113794638843290884, 9.722133654477880552358941816178
