L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 13-s + 15-s + 4·17-s − 4·19-s − 21-s − 23-s − 4·25-s + 27-s − 29-s − 10·31-s − 35-s − 5·37-s − 39-s + 7·41-s + 3·43-s + 45-s − 3·47-s + 49-s + 4·51-s − 4·57-s − 6·59-s − 6·61-s − 63-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 0.970·17-s − 0.917·19-s − 0.218·21-s − 0.208·23-s − 4/5·25-s + 0.192·27-s − 0.185·29-s − 1.79·31-s − 0.169·35-s − 0.821·37-s − 0.160·39-s + 1.09·41-s + 0.457·43-s + 0.149·45-s − 0.437·47-s + 1/7·49-s + 0.560·51-s − 0.529·57-s − 0.781·59-s − 0.768·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.51889348287987392539116933126, −6.97112912634855260726731800801, −5.98282625553545768787394416634, −5.62471569925703895072251295982, −4.58352412874031253265369616006, −3.82148339089185585446052302704, −3.12241666338060391253232806431, −2.22512798505019826946950210143, −1.49671062267355838887576722447, 0,
1.49671062267355838887576722447, 2.22512798505019826946950210143, 3.12241666338060391253232806431, 3.82148339089185585446052302704, 4.58352412874031253265369616006, 5.62471569925703895072251295982, 5.98282625553545768787394416634, 6.97112912634855260726731800801, 7.51889348287987392539116933126