| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s + 9-s − 2·10-s − 4·11-s + 12-s + 13-s + 2·15-s + 16-s + 6·17-s − 18-s − 4·19-s + 2·20-s + 4·22-s − 24-s − 25-s − 26-s + 27-s − 4·29-s − 2·30-s + 4·31-s − 32-s − 4·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s + 0.277·13-s + 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.852·22-s − 0.204·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.742·29-s − 0.365·30-s + 0.718·31-s − 0.176·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.975883561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.975883561\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−8.438743761478596966794413199618, −7.85424941685172240585928204472, −7.34573186581010648982526735653, −6.15921416802721365887103302857, −5.79184364247809877004868875280, −4.77717087937453014939027545576, −3.64348129457895306734574954830, −2.64988242032899518945171767362, −2.07954694349087778477557858666, −0.887963282450702501577518528626,
0.887963282450702501577518528626, 2.07954694349087778477557858666, 2.64988242032899518945171767362, 3.64348129457895306734574954830, 4.77717087937453014939027545576, 5.79184364247809877004868875280, 6.15921416802721365887103302857, 7.34573186581010648982526735653, 7.85424941685172240585928204472, 8.438743761478596966794413199618
