| L(s) = 1 | − 2-s + 3-s + 4-s + 4·5-s − 6-s − 8-s + 9-s − 4·10-s + 12-s + 6·13-s + 4·15-s + 16-s + 17-s − 18-s − 4·19-s + 4·20-s + 6·23-s − 24-s + 11·25-s − 6·26-s + 27-s − 4·29-s − 4·30-s + 6·31-s − 32-s − 34-s + 36-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 + 0.288·12-s + 1.66·13-s + 1.03·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.894·20-s + 1.25·23-s − 0.204·24-s + 11/5·25-s − 1.17·26-s + 0.192·27-s − 0.742·29-s − 0.730·30-s + 1.07·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.944883832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.944883832\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.511277482343380670875082616936, −7.67080076534670349964439281106, −6.65609487669640152451998833504, −6.25650993618347018284029221741, −5.58930669699156824874707798240, −4.60406618583419097735195221715, −3.41650850636158117457558100754, −2.65274646328445039923916322365, −1.75668697624836907999103677619, −1.13374758634790644463533329227,
1.13374758634790644463533329227, 1.75668697624836907999103677619, 2.65274646328445039923916322365, 3.41650850636158117457558100754, 4.60406618583419097735195221715, 5.58930669699156824874707798240, 6.25650993618347018284029221741, 6.65609487669640152451998833504, 7.67080076534670349964439281106, 8.511277482343380670875082616936
