L(s) = 1 | + 2-s + 3-s + 4-s − 4·5-s + 6-s + 8-s + 9-s − 4·10-s − 11-s + 12-s + 13-s − 4·15-s + 16-s − 7·17-s + 18-s − 19-s − 4·20-s − 22-s + 23-s + 24-s + 11·25-s + 26-s + 27-s + 5·29-s − 4·30-s + 5·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.301·11-s + 0.288·12-s + 0.277·13-s − 1.03·15-s + 1/4·16-s − 1.69·17-s + 0.235·18-s − 0.229·19-s − 0.894·20-s − 0.213·22-s + 0.208·23-s + 0.204·24-s + 11/5·25-s + 0.196·26-s + 0.192·27-s + 0.928·29-s − 0.730·30-s + 0.898·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74382 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74382 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.35458443140979, −13.64286982186607, −13.49543057799542, −12.81989980836725, −12.32954534561242, −11.85424395272269, −11.51085231860819, −10.75459401993854, −10.68407865416396, −9.858460846982801, −9.027931614615674, −8.498142393562573, −8.316383027977177, −7.608560254609926, −7.170077458555795, −6.594033570849973, −6.232885466063671, −5.095092154546086, −4.693326859152969, −4.274585360729133, −3.699945471284740, −3.170039784379684, −2.623829892715224, −1.883928786054468, −0.8698263027695128, 0,
0.8698263027695128, 1.883928786054468, 2.623829892715224, 3.170039784379684, 3.699945471284740, 4.274585360729133, 4.693326859152969, 5.095092154546086, 6.232885466063671, 6.594033570849973, 7.170077458555795, 7.608560254609926, 8.316383027977177, 8.498142393562573, 9.027931614615674, 9.858460846982801, 10.68407865416396, 10.75459401993854, 11.51085231860819, 11.85424395272269, 12.32954534561242, 12.81989980836725, 13.49543057799542, 13.64286982186607, 14.35458443140979