L(s) = 1 | + 7-s − 5·11-s + 6·13-s + 4·17-s + 6·19-s − 3·23-s + 3·29-s − 2·31-s − 7·37-s + 4·41-s − 7·43-s + 2·47-s + 49-s − 10·53-s − 14·59-s + 4·61-s + 3·67-s − 13·71-s − 16·73-s − 5·77-s − 79-s − 10·83-s − 10·89-s + 6·91-s + 2·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 1.50·11-s + 1.66·13-s + 0.970·17-s + 1.37·19-s − 0.625·23-s + 0.557·29-s − 0.359·31-s − 1.15·37-s + 0.624·41-s − 1.06·43-s + 0.291·47-s + 1/7·49-s − 1.37·53-s − 1.82·59-s + 0.512·61-s + 0.366·67-s − 1.54·71-s − 1.87·73-s − 0.569·77-s − 0.112·79-s − 1.09·83-s − 1.05·89-s + 0.628·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.60282636496830, −15.35807717628532, −14.33422115414118, −14.03364793809184, −13.56699606494820, −12.94713186956098, −12.49831370693870, −11.67717613071798, −11.43396180819035, −10.50269035779017, −10.43241562148860, −9.681245295317383, −8.946135507105619, −8.375721537244806, −7.836387065406678, −7.486869541131486, −6.645658797205256, −5.793830823345929, −5.557789231065007, −4.853459538540935, −4.091671406236172, −3.196157076232073, −2.964035702457920, −1.724151974035154, −1.194310704883346, 0,
1.194310704883346, 1.724151974035154, 2.964035702457920, 3.196157076232073, 4.091671406236172, 4.853459538540935, 5.557789231065007, 5.793830823345929, 6.645658797205256, 7.486869541131486, 7.836387065406678, 8.375721537244806, 8.946135507105619, 9.681245295317383, 10.43241562148860, 10.50269035779017, 11.43396180819035, 11.67717613071798, 12.49831370693870, 12.94713186956098, 13.56699606494820, 14.03364793809184, 14.33422115414118, 15.35807717628532, 15.60282636496830