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
−7.21372180010215654252745918078, −6.79412967415503416242406801210, −5.86541918866092582633004991010, −5.10274185521135129723767956032, −4.69508036462649347825894031805, −4.00209041095410698766008005561, −2.88734375994309713503561276188, −2.44138305098421678240160356990, −1.43566032589665524067972287384, 0,
1.43566032589665524067972287384, 2.44138305098421678240160356990, 2.88734375994309713503561276188, 4.00209041095410698766008005561, 4.69508036462649347825894031805, 5.10274185521135129723767956032, 5.86541918866092582633004991010, 6.79412967415503416242406801210, 7.21372180010215654252745918078