| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s − 3·8-s + 9-s + 10-s − 4·11-s − 12-s + 2·13-s + 15-s − 16-s + 2·17-s + 18-s − 20-s − 4·22-s − 3·24-s + 25-s + 2·26-s + 27-s + 2·29-s + 30-s + 5·32-s − 4·33-s + 2·34-s − 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.223·20-s − 0.852·22-s − 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.371·29-s + 0.182·30-s + 0.883·32-s − 0.696·33-s + 0.342·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.929624646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.929624646\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.179427569545746398797779230432, −7.60719004255564017514118173943, −6.56470169945492021349835720384, −5.84128873133936523436206316749, −5.21484715703970999635976715829, −4.52985320583773135666017839183, −3.66793113681489031905327880745, −2.96337185591461309202736907725, −2.19847936643903163153856477125, −0.803857922530386741108529528003,
0.803857922530386741108529528003, 2.19847936643903163153856477125, 2.96337185591461309202736907725, 3.66793113681489031905327880745, 4.52985320583773135666017839183, 5.21484715703970999635976715829, 5.84128873133936523436206316749, 6.56470169945492021349835720384, 7.60719004255564017514118173943, 8.179427569545746398797779230432
