| L(s) = 1 | − 2·2-s + 4·4-s − 16·5-s + 7·7-s − 8·8-s + 32·10-s + 15·11-s − 13·13-s − 14·14-s + 16·16-s + 44·17-s − 138·19-s − 64·20-s − 30·22-s − 111·23-s + 131·25-s + 26·26-s + 28·28-s + 12·29-s + 215·31-s − 32·32-s − 88·34-s − 112·35-s + 55·37-s + 276·38-s + 128·40-s + 133·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.43·5-s + 0.377·7-s − 0.353·8-s + 1.01·10-s + 0.411·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.627·17-s − 1.66·19-s − 0.715·20-s − 0.290·22-s − 1.00·23-s + 1.04·25-s + 0.196·26-s + 0.188·28-s + 0.0768·29-s + 1.24·31-s − 0.176·32-s − 0.443·34-s − 0.540·35-s + 0.244·37-s + 1.17·38-s + 0.505·40-s + 0.506·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7010842840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7010842840\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 15 T + p^{3} T^{2} \) |
| 17 | \( 1 - 44 T + p^{3} T^{2} \) |
| 19 | \( 1 + 138 T + p^{3} T^{2} \) |
| 23 | \( 1 + 111 T + p^{3} T^{2} \) |
| 29 | \( 1 - 12 T + p^{3} T^{2} \) |
| 31 | \( 1 - 215 T + p^{3} T^{2} \) |
| 37 | \( 1 - 55 T + p^{3} T^{2} \) |
| 41 | \( 1 - 133 T + p^{3} T^{2} \) |
| 43 | \( 1 + 180 T + p^{3} T^{2} \) |
| 47 | \( 1 + 471 T + p^{3} T^{2} \) |
| 53 | \( 1 - 260 T + p^{3} T^{2} \) |
| 59 | \( 1 + 110 T + p^{3} T^{2} \) |
| 61 | \( 1 + 271 T + p^{3} T^{2} \) |
| 67 | \( 1 + 799 T + p^{3} T^{2} \) |
| 71 | \( 1 + 912 T + p^{3} T^{2} \) |
| 73 | \( 1 - 747 T + p^{3} T^{2} \) |
| 79 | \( 1 + 883 T + p^{3} T^{2} \) |
| 83 | \( 1 - 924 T + p^{3} T^{2} \) |
| 89 | \( 1 + 142 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1407 T + p^{3} T^{2} \) |
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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
−8.770172360604791291426472356273, −8.178933818499922630761520152990, −7.71189482262691876603598088640, −6.80353327916831744725236952176, −6.00709845143487144263813083201, −4.63762758612087171070795819436, −4.01776319857391146633099511749, −2.94861676205753297236564129416, −1.70823289990442637200619136167, −0.44621338207884400848971089720,
0.44621338207884400848971089720, 1.70823289990442637200619136167, 2.94861676205753297236564129416, 4.01776319857391146633099511749, 4.63762758612087171070795819436, 6.00709845143487144263813083201, 6.80353327916831744725236952176, 7.71189482262691876603598088640, 8.178933818499922630761520152990, 8.770172360604791291426472356273
