L(s) = 1 | + 2-s + 4-s + 2·5-s − 7-s + 8-s + 2·10-s + 4·13-s − 14-s + 16-s + 6·19-s + 2·20-s + 23-s − 25-s + 4·26-s − 28-s + 6·29-s − 10·31-s + 32-s − 2·35-s − 6·37-s + 6·38-s + 2·40-s + 2·41-s + 12·43-s + 46-s − 10·47-s + 49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.37·19-s + 0.447·20-s + 0.208·23-s − 1/5·25-s + 0.784·26-s − 0.188·28-s + 1.11·29-s − 1.79·31-s + 0.176·32-s − 0.338·35-s − 0.986·37-s + 0.973·38-s + 0.316·40-s + 0.312·41-s + 1.82·43-s + 0.147·46-s − 1.45·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.656958548\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.656958548\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.936732363586290011899211377258, −7.88098461725684339177736964361, −7.07315686030041310759890000678, −6.30281053414804727332588841869, −5.67091239465789003054096702349, −5.08457267381911005513811491134, −3.89490884964271640465660241639, −3.23531582861158044561746026836, −2.19691714145171113375489744188, −1.14894064998588642402299361358,
1.14894064998588642402299361358, 2.19691714145171113375489744188, 3.23531582861158044561746026836, 3.89490884964271640465660241639, 5.08457267381911005513811491134, 5.67091239465789003054096702349, 6.30281053414804727332588841869, 7.07315686030041310759890000678, 7.88098461725684339177736964361, 8.936732363586290011899211377258