L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 7·13-s + 14-s + 16-s + 3·17-s − 19-s − 6·23-s − 7·26-s − 28-s + 3·29-s − 4·31-s − 32-s − 3·34-s − 2·37-s + 38-s − 6·41-s − 2·43-s + 6·46-s − 3·47-s + 49-s + 7·52-s − 9·53-s + 56-s − 3·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.94·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s − 1.25·23-s − 1.37·26-s − 0.188·28-s + 0.557·29-s − 0.718·31-s − 0.176·32-s − 0.514·34-s − 0.328·37-s + 0.162·38-s − 0.937·41-s − 0.304·43-s + 0.884·46-s − 0.437·47-s + 1/7·49-s + 0.970·52-s − 1.23·53-s + 0.133·56-s − 0.393·58-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 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.53041167487383458690586078852, −6.58359631318036486251927856719, −6.16486287396425270847027560315, −5.57238817327691904494992114182, −4.49546486898816397402479259668, −3.57507327494934928445423874483, −3.15694455227798503250851757647, −1.90041591950880760450458273333, −1.23492201308239163168232755810, 0,
1.23492201308239163168232755810, 1.90041591950880760450458273333, 3.15694455227798503250851757647, 3.57507327494934928445423874483, 4.49546486898816397402479259668, 5.57238817327691904494992114182, 6.16486287396425270847027560315, 6.58359631318036486251927856719, 7.53041167487383458690586078852