| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 2·11-s − 6·13-s − 14-s + 16-s − 6·17-s + 19-s − 2·22-s − 3·23-s − 6·26-s − 28-s + 2·29-s − 10·31-s + 32-s − 6·34-s − 2·37-s + 38-s − 9·41-s − 8·43-s − 2·44-s − 3·46-s − 8·47-s − 6·49-s − 6·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.603·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.229·19-s − 0.426·22-s − 0.625·23-s − 1.17·26-s − 0.188·28-s + 0.371·29-s − 1.79·31-s + 0.176·32-s − 1.02·34-s − 0.328·37-s + 0.162·38-s − 1.40·41-s − 1.21·43-s − 0.301·44-s − 0.442·46-s − 1.16·47-s − 6/7·49-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.59589706919971, −14.11064370482278, −13.35566854887071, −13.15907409184159, −12.68548775485149, −12.06303068262703, −11.77524328565640, −11.03400540923899, −10.71984841541293, −9.957638889455200, −9.673870457485503, −9.117958733158608, −8.243739681545720, −7.979407493641288, −7.184443285348683, −6.749306918131351, −6.461281806219603, −5.467301845299557, −5.157568112869882, −4.675663769152627, −4.026790215006321, −3.293216385527061, −2.824898390587228, −2.059293362723691, −1.685647878575854, 0, 0,
1.685647878575854, 2.059293362723691, 2.824898390587228, 3.293216385527061, 4.026790215006321, 4.675663769152627, 5.157568112869882, 5.467301845299557, 6.461281806219603, 6.749306918131351, 7.184443285348683, 7.979407493641288, 8.243739681545720, 9.117958733158608, 9.673870457485503, 9.957638889455200, 10.71984841541293, 11.03400540923899, 11.77524328565640, 12.06303068262703, 12.68548775485149, 13.15907409184159, 13.35566854887071, 14.11064370482278, 14.59589706919971
