L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 4·11-s + 12-s − 13-s + 16-s − 18-s + 19-s − 4·22-s + 4·23-s − 24-s + 26-s + 27-s − 3·29-s + 4·31-s − 32-s + 4·33-s + 36-s − 5·37-s − 38-s − 39-s + 9·41-s + 2·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 0.277·13-s + 1/4·16-s − 0.235·18-s + 0.229·19-s − 0.852·22-s + 0.834·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.557·29-s + 0.718·31-s − 0.176·32-s + 0.696·33-s + 1/6·36-s − 0.821·37-s − 0.162·38-s − 0.160·39-s + 1.40·41-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.708728428\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.708728428\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.246022407609428093653317779340, −8.514158509207057194140551497649, −7.72700124301635413334465383184, −6.96185452609991083183049607731, −6.31169459943465094676423661209, −5.16279690910238322692877336497, −4.06813422357611582515536400915, −3.18234455028814944138477269225, −2.10438601929059891449152955112, −0.996110380048603486177702844541,
0.996110380048603486177702844541, 2.10438601929059891449152955112, 3.18234455028814944138477269225, 4.06813422357611582515536400915, 5.16279690910238322692877336497, 6.31169459943465094676423661209, 6.96185452609991083183049607731, 7.72700124301635413334465383184, 8.514158509207057194140551497649, 9.246022407609428093653317779340