| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 4·11-s + 2·13-s + 14-s + 16-s − 2·17-s − 20-s − 4·22-s + 8·23-s + 25-s + 2·26-s + 28-s + 6·29-s + 8·31-s + 32-s − 2·34-s − 35-s + 2·37-s − 40-s + 2·41-s − 12·43-s − 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.169·35-s + 0.328·37-s − 0.158·40-s + 0.312·41-s − 1.82·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.19178338210166, −12.86823068458375, −12.18881707049595, −11.80152457869026, −11.45688017251627, −10.75742922343316, −10.62198288162279, −10.14178412968821, −9.428445660382907, −8.806361699777899, −8.303535134005230, −8.097362179009306, −7.395361150115569, −6.897016399019519, −6.587645813786041, −5.844349007840462, −5.362780109554719, −4.916095846623809, −4.410680499177293, −4.048686025699837, −3.148232093855754, −2.827298968766981, −2.412571788105576, −1.421569940687072, −0.9495682615918058, 0,
0.9495682615918058, 1.421569940687072, 2.412571788105576, 2.827298968766981, 3.148232093855754, 4.048686025699837, 4.410680499177293, 4.916095846623809, 5.362780109554719, 5.844349007840462, 6.587645813786041, 6.897016399019519, 7.395361150115569, 8.097362179009306, 8.303535134005230, 8.806361699777899, 9.428445660382907, 10.14178412968821, 10.62198288162279, 10.75742922343316, 11.45688017251627, 11.80152457869026, 12.18881707049595, 12.86823068458375, 13.19178338210166
