| L(s) = 1 | + 3-s − 5-s + 7-s − 2·9-s + 11-s − 4·13-s − 15-s − 6·17-s + 21-s + 2·23-s + 25-s − 5·27-s + 10·29-s − 6·31-s + 33-s − 35-s − 4·39-s + 9·41-s − 4·43-s + 2·45-s − 12·47-s + 49-s − 6·51-s + 4·53-s − 55-s − 3·59-s − 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.301·11-s − 1.10·13-s − 0.258·15-s − 1.45·17-s + 0.218·21-s + 0.417·23-s + 1/5·25-s − 0.962·27-s + 1.85·29-s − 1.07·31-s + 0.174·33-s − 0.169·35-s − 0.640·39-s + 1.40·41-s − 0.609·43-s + 0.298·45-s − 1.75·47-s + 1/7·49-s − 0.840·51-s + 0.549·53-s − 0.134·55-s − 0.390·59-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−13.22060667372620, −12.82310115703163, −12.43450974614610, −11.75120550426560, −11.35288249097507, −11.20086039888356, −10.43166604898660, −9.992443153239382, −9.395683190397994, −8.934509818221604, −8.566015396803031, −8.167186193249950, −7.569608066463162, −7.159539192052649, −6.602479476472014, −6.158499919913974, −5.380327841387129, −4.899859573942523, −4.453376746219759, −3.949883382555461, −3.195249544121483, −2.772373496894787, −2.236368519564396, −1.666458233178595, −0.7017270483136128, 0,
0.7017270483136128, 1.666458233178595, 2.236368519564396, 2.772373496894787, 3.195249544121483, 3.949883382555461, 4.453376746219759, 4.899859573942523, 5.380327841387129, 6.158499919913974, 6.602479476472014, 7.159539192052649, 7.569608066463162, 8.167186193249950, 8.566015396803031, 8.934509818221604, 9.395683190397994, 9.992443153239382, 10.43166604898660, 11.20086039888356, 11.35288249097507, 11.75120550426560, 12.43450974614610, 12.82310115703163, 13.22060667372620
