L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 12-s − 6·13-s + 14-s + 16-s − 4·17-s + 18-s − 6·19-s + 21-s − 4·23-s + 24-s − 5·25-s − 6·26-s + 27-s + 28-s − 6·29-s − 2·31-s + 32-s − 4·34-s + 36-s + 10·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 1.37·19-s + 0.218·21-s − 0.834·23-s + 0.204·24-s − 25-s − 1.17·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s − 0.359·31-s + 0.176·32-s − 0.685·34-s + 1/6·36-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5082 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5082 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.88044768145892684250248521093, −7.15083214327735671821742085196, −6.42153264848366358553087579074, −5.64719531132362490687226713521, −4.63059859799734414629334650532, −4.33149302139153054926791784615, −3.35278591108779725059175891001, −2.22064810010506599000527056391, −2.00017416871630464571063850965, 0,
2.00017416871630464571063850965, 2.22064810010506599000527056391, 3.35278591108779725059175891001, 4.33149302139153054926791784615, 4.63059859799734414629334650532, 5.64719531132362490687226713521, 6.42153264848366358553087579074, 7.15083214327735671821742085196, 7.88044768145892684250248521093