L(s) = 1 | + 3·3-s − 5·5-s − 4·7-s + 9·9-s − 40·11-s + 90·13-s − 15·15-s − 70·17-s − 40·19-s − 12·21-s + 108·23-s + 25·25-s + 27·27-s − 166·29-s − 40·31-s − 120·33-s + 20·35-s + 130·37-s + 270·39-s − 310·41-s + 268·43-s − 45·45-s − 556·47-s − 327·49-s − 210·51-s + 370·53-s + 200·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.215·7-s + 1/3·9-s − 1.09·11-s + 1.92·13-s − 0.258·15-s − 0.998·17-s − 0.482·19-s − 0.124·21-s + 0.979·23-s + 1/5·25-s + 0.192·27-s − 1.06·29-s − 0.231·31-s − 0.633·33-s + 0.0965·35-s + 0.577·37-s + 1.10·39-s − 1.18·41-s + 0.950·43-s − 0.149·45-s − 1.72·47-s − 0.953·49-s − 0.576·51-s + 0.958·53-s + 0.490·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
good | 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 - 90 T + p^{3} T^{2} \) |
| 17 | \( 1 + 70 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 - 108 T + p^{3} T^{2} \) |
| 29 | \( 1 + 166 T + p^{3} T^{2} \) |
| 31 | \( 1 + 40 T + p^{3} T^{2} \) |
| 37 | \( 1 - 130 T + p^{3} T^{2} \) |
| 41 | \( 1 + 310 T + p^{3} T^{2} \) |
| 43 | \( 1 - 268 T + p^{3} T^{2} \) |
| 47 | \( 1 + 556 T + p^{3} T^{2} \) |
| 53 | \( 1 - 370 T + p^{3} T^{2} \) |
| 59 | \( 1 + 240 T + p^{3} T^{2} \) |
| 61 | \( 1 - 130 T + p^{3} T^{2} \) |
| 67 | \( 1 + 876 T + p^{3} T^{2} \) |
| 71 | \( 1 + 840 T + p^{3} T^{2} \) |
| 73 | \( 1 - 250 T + p^{3} T^{2} \) |
| 79 | \( 1 + 880 T + p^{3} T^{2} \) |
| 83 | \( 1 - 188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 726 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1550 T + p^{3} T^{2} \) |
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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
−8.945615649510797696707761954836, −8.522547823766320632033095461657, −7.66250472318992918521744703521, −6.74464722362504498654562508597, −5.82416685321950769710624579277, −4.64679331583814750684961567080, −3.69347220426552342538217971033, −2.83709946597990100342387899398, −1.53596338138111763827177591256, 0,
1.53596338138111763827177591256, 2.83709946597990100342387899398, 3.69347220426552342538217971033, 4.64679331583814750684961567080, 5.82416685321950769710624579277, 6.74464722362504498654562508597, 7.66250472318992918521744703521, 8.522547823766320632033095461657, 8.945615649510797696707761954836