L(s) = 1 | + 2-s + 4-s + 2·5-s − 7-s + 8-s − 3·9-s + 2·10-s + 4·11-s − 14-s + 16-s − 6·17-s − 3·18-s + 4·19-s + 2·20-s + 4·22-s + 23-s − 25-s − 28-s + 29-s + 2·31-s + 32-s − 6·34-s − 2·35-s − 3·36-s − 6·37-s + 4·38-s + 2·40-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s − 9-s + 0.632·10-s + 1.20·11-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s + 0.447·20-s + 0.852·22-s + 0.208·23-s − 1/5·25-s − 0.188·28-s + 0.185·29-s + 0.359·31-s + 0.176·32-s − 1.02·34-s − 0.338·35-s − 1/2·36-s − 0.986·37-s + 0.648·38-s + 0.316·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.808915314\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.808915314\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.51425491533097242059631944392, −6.67373066206834922506626506588, −6.36598032253970446568235683607, −5.63977904181894921063027581850, −5.10076946841530238582195331413, −4.13929415057477057427871613749, −3.51458354287906293771444004523, −2.58550631530738384728845364071, −2.02366965845819940676151290072, −0.835852958601903188105758811051,
0.835852958601903188105758811051, 2.02366965845819940676151290072, 2.58550631530738384728845364071, 3.51458354287906293771444004523, 4.13929415057477057427871613749, 5.10076946841530238582195331413, 5.63977904181894921063027581850, 6.36598032253970446568235683607, 6.67373066206834922506626506588, 7.51425491533097242059631944392