| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 4·11-s + 12-s + 13-s + 16-s + 2·17-s + 18-s + 19-s − 4·22-s − 2·23-s + 24-s + 26-s + 27-s + 4·29-s + 32-s − 4·33-s + 2·34-s + 36-s + 3·37-s + 38-s + 39-s + 12·41-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 + 0.485·17-s + 0.235·18-s + 0.229·19-s − 0.852·22-s − 0.417·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.742·29-s + 0.176·32-s − 0.696·33-s + 0.342·34-s + 1/6·36-s + 0.493·37-s + 0.162·38-s + 0.160·39-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.081373054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.081373054\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−7.74967791095177135628187865458, −7.40422970878670391796200311605, −6.35750716781920291052263498178, −5.79389454254838646126223600523, −5.00219652450133416208496186198, −4.33419657596198654516199421193, −3.50328929010646669042707524225, −2.77196452272813922944624280433, −2.14050419303078085627220260630, −0.893213351326383350428553648890,
0.893213351326383350428553648890, 2.14050419303078085627220260630, 2.77196452272813922944624280433, 3.50328929010646669042707524225, 4.33419657596198654516199421193, 5.00219652450133416208496186198, 5.79389454254838646126223600523, 6.35750716781920291052263498178, 7.40422970878670391796200311605, 7.74967791095177135628187865458
