| L(s) = 1 | + 1.33·2-s − 6.23·4-s − 10.6·7-s − 18.9·8-s − 11.2·11-s + 2.74·13-s − 14.2·14-s + 24.6·16-s + 29.5·17-s − 31.1·19-s − 14.9·22-s − 116.·23-s + 3.64·26-s + 66.6·28-s − 108.·29-s + 70.7·31-s + 184.·32-s + 39.3·34-s + 282.·37-s − 41.3·38-s + 425.·41-s + 312.·43-s + 70.1·44-s − 155.·46-s + 193.·47-s − 228.·49-s − 17.0·52-s + ⋯ |
| L(s) = 1 | + 0.470·2-s − 0.778·4-s − 0.577·7-s − 0.836·8-s − 0.308·11-s + 0.0584·13-s − 0.271·14-s + 0.385·16-s + 0.421·17-s − 0.375·19-s − 0.145·22-s − 1.06·23-s + 0.0274·26-s + 0.449·28-s − 0.694·29-s + 0.410·31-s + 1.01·32-s + 0.198·34-s + 1.25·37-s − 0.176·38-s + 1.62·41-s + 1.10·43-s + 0.240·44-s − 0.498·46-s + 0.600·47-s − 0.666·49-s − 0.0455·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.462388122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.462388122\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.33T + 8T^{2} \) |
| 7 | \( 1 + 10.6T + 343T^{2} \) |
| 11 | \( 1 + 11.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.74T + 2.19e3T^{2} \) |
| 17 | \( 1 - 29.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 70.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 282.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 425.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 312.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 193.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 103.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 494.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 424.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 586.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 302.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 525.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.42e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 25.6T + 9.12e5T^{2} \) |
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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
−9.906359841846909748790424644669, −9.387053790166472625314701388336, −8.380103591457539942018723563561, −7.55360738512906989413465302895, −6.23072750855290294943568944255, −5.62064853562140995481009774597, −4.45536592069574156669927465954, −3.67542234591668782930673174128, −2.50477401194794527304551889057, −0.64435053794068204780847327356,
0.64435053794068204780847327356, 2.50477401194794527304551889057, 3.67542234591668782930673174128, 4.45536592069574156669927465954, 5.62064853562140995481009774597, 6.23072750855290294943568944255, 7.55360738512906989413465302895, 8.380103591457539942018723563561, 9.387053790166472625314701388336, 9.906359841846909748790424644669
