| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 6·11-s + 12-s − 13-s − 14-s + 16-s + 3·17-s − 18-s − 4·19-s + 21-s − 6·22-s − 3·23-s − 24-s + 26-s + 27-s + 28-s + 3·29-s + 5·31-s − 32-s + 6·33-s − 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.917·19-s + 0.218·21-s − 1.27·22-s − 0.625·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 0.557·29-s + 0.898·31-s − 0.176·32-s + 1.04·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.648484523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.648484523\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.902052486220863088026891256086, −8.852617318380859459119284920470, −8.565377541517557168162281813646, −7.50271823062817453512163835510, −6.75966491208539742900986742997, −5.88196315142558299370134897673, −4.45784300402311455229373043929, −3.59383686991635819010142129297, −2.27258262554885043386104968494, −1.17771918598491916536579617003,
1.17771918598491916536579617003, 2.27258262554885043386104968494, 3.59383686991635819010142129297, 4.45784300402311455229373043929, 5.88196315142558299370134897673, 6.75966491208539742900986742997, 7.50271823062817453512163835510, 8.565377541517557168162281813646, 8.852617318380859459119284920470, 9.902052486220863088026891256086
