L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 4·11-s − 6·13-s + 14-s + 16-s − 4·17-s − 4·22-s − 4·23-s + 6·26-s − 28-s − 3·29-s + 7·31-s − 32-s + 4·34-s + 37-s + 7·41-s + 10·43-s + 4·44-s + 4·46-s + 13·47-s + 49-s − 6·52-s − 6·53-s + 56-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.20·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.852·22-s − 0.834·23-s + 1.17·26-s − 0.188·28-s − 0.557·29-s + 1.25·31-s − 0.176·32-s + 0.685·34-s + 0.164·37-s + 1.09·41-s + 1.52·43-s + 0.603·44-s + 0.589·46-s + 1.89·47-s + 1/7·49-s − 0.832·52-s − 0.824·53-s + 0.133·56-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 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.44938209328880730387539177172, −6.71565645067984072568496716858, −6.24862150302888314053698385583, −5.42627756526030903880835373264, −4.39590641910139547691191321645, −3.94819450407876239546169643226, −2.68665003226096656161243988692, −2.25726713756267008942843236200, −1.08171226124613748582138172247, 0,
1.08171226124613748582138172247, 2.25726713756267008942843236200, 2.68665003226096656161243988692, 3.94819450407876239546169643226, 4.39590641910139547691191321645, 5.42627756526030903880835373264, 6.24862150302888314053698385583, 6.71565645067984072568496716858, 7.44938209328880730387539177172