L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 11-s − 5·13-s − 14-s + 16-s − 4·17-s + 7·19-s − 22-s − 5·26-s − 28-s + 6·29-s + 32-s − 4·34-s + 4·37-s + 7·38-s + 3·41-s − 11·43-s − 44-s + 47-s + 49-s − 5·52-s + 53-s − 56-s + 6·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.301·11-s − 1.38·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s + 1.60·19-s − 0.213·22-s − 0.980·26-s − 0.188·28-s + 1.11·29-s + 0.176·32-s − 0.685·34-s + 0.657·37-s + 1.13·38-s + 0.468·41-s − 1.67·43-s − 0.150·44-s + 0.145·47-s + 1/7·49-s − 0.693·52-s + 0.137·53-s − 0.133·56-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.36996431792706569040468028762, −6.60454505395403254518614640603, −5.99370621499675214476176657766, −5.05984547288473290372516999634, −4.78052536992126218301071767405, −3.86377607536591822886185150852, −2.90814819344513500234870477461, −2.54623649659736378575348422722, −1.35785375005089063654445415145, 0,
1.35785375005089063654445415145, 2.54623649659736378575348422722, 2.90814819344513500234870477461, 3.86377607536591822886185150852, 4.78052536992126218301071767405, 5.05984547288473290372516999634, 5.99370621499675214476176657766, 6.60454505395403254518614640603, 7.36996431792706569040468028762