| L(s) = 1 | − 2·2-s − 4·4-s + 24·8-s − 10·11-s + 80·13-s − 16·16-s + 7·17-s − 113·19-s + 20·22-s − 81·23-s − 160·26-s + 220·29-s − 189·31-s − 160·32-s − 14·34-s − 170·37-s + 226·38-s + 130·41-s − 10·43-s + 40·44-s + 162·46-s + 160·47-s − 343·49-s − 320·52-s + 631·53-s − 440·58-s + 560·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 0.274·11-s + 1.70·13-s − 1/4·16-s + 0.0998·17-s − 1.36·19-s + 0.193·22-s − 0.734·23-s − 1.20·26-s + 1.40·29-s − 1.09·31-s − 0.883·32-s − 0.0706·34-s − 0.755·37-s + 0.964·38-s + 0.495·41-s − 0.0354·43-s + 0.137·44-s + 0.519·46-s + 0.496·47-s − 49-s − 0.853·52-s + 1.63·53-s − 0.996·58-s + 1.23·59-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
| \(L(2)\) |
\(\approx\) |
\(1.051549955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.051549955\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p T + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + 10 T + p^{3} T^{2} \) |
| 13 | \( 1 - 80 T + p^{3} T^{2} \) |
| 17 | \( 1 - 7 T + p^{3} T^{2} \) |
| 19 | \( 1 + 113 T + p^{3} T^{2} \) |
| 23 | \( 1 + 81 T + p^{3} T^{2} \) |
| 29 | \( 1 - 220 T + p^{3} T^{2} \) |
| 31 | \( 1 + 189 T + p^{3} T^{2} \) |
| 37 | \( 1 + 170 T + p^{3} T^{2} \) |
| 41 | \( 1 - 130 T + p^{3} T^{2} \) |
| 43 | \( 1 + 10 T + p^{3} T^{2} \) |
| 47 | \( 1 - 160 T + p^{3} T^{2} \) |
| 53 | \( 1 - 631 T + p^{3} T^{2} \) |
| 59 | \( 1 - 560 T + p^{3} T^{2} \) |
| 61 | \( 1 - 229 T + p^{3} T^{2} \) |
| 67 | \( 1 + 750 T + p^{3} T^{2} \) |
| 71 | \( 1 + 890 T + p^{3} T^{2} \) |
| 73 | \( 1 - 890 T + p^{3} T^{2} \) |
| 79 | \( 1 + 27 T + p^{3} T^{2} \) |
| 83 | \( 1 - 429 T + p^{3} T^{2} \) |
| 89 | \( 1 - 750 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1480 T + p^{3} T^{2} \) |
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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
−10.25568621395669623652478502475, −8.959537146632473059915544997972, −8.588775636096517273164333831876, −7.77538129577762629732119764149, −6.61395261131403054882410943788, −5.66590465396959649829300645676, −4.45598214351314180131669587651, −3.60485700822879633842008529539, −1.92556283558731529427559757737, −0.67967671919860592609924455428,
0.67967671919860592609924455428, 1.92556283558731529427559757737, 3.60485700822879633842008529539, 4.45598214351314180131669587651, 5.66590465396959649829300645676, 6.61395261131403054882410943788, 7.77538129577762629732119764149, 8.588775636096517273164333831876, 8.959537146632473059915544997972, 10.25568621395669623652478502475
