L(s) = 1 | + 4·5-s − 7-s + 2·11-s + 6·13-s + 4·17-s + 4·19-s − 2·23-s + 11·25-s − 2·29-s − 4·35-s − 2·37-s + 4·43-s − 12·47-s + 49-s − 6·53-s + 8·55-s − 8·59-s − 6·61-s + 24·65-s + 8·67-s − 14·71-s − 2·73-s − 2·77-s + 12·79-s − 4·83-s + 16·85-s − 6·91-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 0.377·7-s + 0.603·11-s + 1.66·13-s + 0.970·17-s + 0.917·19-s − 0.417·23-s + 11/5·25-s − 0.371·29-s − 0.676·35-s − 0.328·37-s + 0.609·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s − 1.04·59-s − 0.768·61-s + 2.97·65-s + 0.977·67-s − 1.66·71-s − 0.234·73-s − 0.227·77-s + 1.35·79-s − 0.439·83-s + 1.73·85-s − 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.210132407\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.210132407\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.629751576240571797976949670652, −7.72604486612696592247209656788, −6.71189699012874428808268514597, −6.09209328369221800215319890295, −5.74755430914602536337090529880, −4.86265553633570999718490113346, −3.63611416585453122580621062957, −3.00487505240709347914613380548, −1.75098525765288305362479637690, −1.17710716193186405280124036592,
1.17710716193186405280124036592, 1.75098525765288305362479637690, 3.00487505240709347914613380548, 3.63611416585453122580621062957, 4.86265553633570999718490113346, 5.74755430914602536337090529880, 6.09209328369221800215319890295, 6.71189699012874428808268514597, 7.72604486612696592247209656788, 8.629751576240571797976949670652