| L(s) = 1 | + 2·2-s + 4·4-s − 14·7-s + 8·8-s + 3·11-s − 47·13-s − 28·14-s + 16·16-s + 39·17-s + 32·19-s + 6·22-s + 99·23-s − 94·26-s − 56·28-s + 51·29-s + 83·31-s + 32·32-s + 78·34-s − 314·37-s + 64·38-s − 108·41-s − 299·43-s + 12·44-s + 198·46-s − 531·47-s − 147·49-s − 188·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.0822·11-s − 1.00·13-s − 0.534·14-s + 1/4·16-s + 0.556·17-s + 0.386·19-s + 0.0581·22-s + 0.897·23-s − 0.709·26-s − 0.377·28-s + 0.326·29-s + 0.480·31-s + 0.176·32-s + 0.393·34-s − 1.39·37-s + 0.273·38-s − 0.411·41-s − 1.06·43-s + 0.0411·44-s + 0.634·46-s − 1.64·47-s − 3/7·49-s − 0.501·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 3 T + p^{3} T^{2} \) |
| 13 | \( 1 + 47 T + p^{3} T^{2} \) |
| 17 | \( 1 - 39 T + p^{3} T^{2} \) |
| 19 | \( 1 - 32 T + p^{3} T^{2} \) |
| 23 | \( 1 - 99 T + p^{3} T^{2} \) |
| 29 | \( 1 - 51 T + p^{3} T^{2} \) |
| 31 | \( 1 - 83 T + p^{3} T^{2} \) |
| 37 | \( 1 + 314 T + p^{3} T^{2} \) |
| 41 | \( 1 + 108 T + p^{3} T^{2} \) |
| 43 | \( 1 + 299 T + p^{3} T^{2} \) |
| 47 | \( 1 + 531 T + p^{3} T^{2} \) |
| 53 | \( 1 + 564 T + p^{3} T^{2} \) |
| 59 | \( 1 - 12 T + p^{3} T^{2} \) |
| 61 | \( 1 - 230 T + p^{3} T^{2} \) |
| 67 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 71 | \( 1 - 120 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1106 T + p^{3} T^{2} \) |
| 79 | \( 1 + 739 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1086 T + p^{3} T^{2} \) |
| 89 | \( 1 + 120 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1642 T + p^{3} T^{2} \) |
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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
−8.866399956819512231931710986603, −7.85629775974044779333710572127, −6.98591666961703626990597599316, −6.40818764460253507737191620262, −5.31002910472011315545124437443, −4.71260857600479750760909332844, −3.44202701952137217735425207239, −2.87369228441141289965083776799, −1.53129210161528466072454122651, 0,
1.53129210161528466072454122651, 2.87369228441141289965083776799, 3.44202701952137217735425207239, 4.71260857600479750760909332844, 5.31002910472011315545124437443, 6.40818764460253507737191620262, 6.98591666961703626990597599316, 7.85629775974044779333710572127, 8.866399956819512231931710986603
