L(s) = 1 | + 2·7-s + 13-s − 2·17-s − 2·19-s − 8·23-s − 5·25-s − 6·29-s + 2·31-s − 6·37-s + 4·43-s − 8·47-s − 3·49-s + 6·53-s − 4·59-s + 2·61-s + 2·67-s − 4·71-s − 2·73-s + 12·79-s + 12·83-s − 12·89-s + 2·91-s − 18·97-s + 2·101-s + 12·103-s + 4·107-s − 2·109-s + ⋯ |
L(s) = 1 | + 0.755·7-s + 0.277·13-s − 0.485·17-s − 0.458·19-s − 1.66·23-s − 25-s − 1.11·29-s + 0.359·31-s − 0.986·37-s + 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.824·53-s − 0.520·59-s + 0.256·61-s + 0.244·67-s − 0.474·71-s − 0.234·73-s + 1.35·79-s + 1.31·83-s − 1.27·89-s + 0.209·91-s − 1.82·97-s + 0.199·101-s + 1.18·103-s + 0.386·107-s − 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.097096554375915957754317563274, −7.59256477164320569305091666261, −6.60083796180378457282385209064, −5.91892391219779025956042628760, −5.13608091937105447697975783959, −4.23894776027283377880109172566, −3.62406434056639048963936774853, −2.28956610059431075582349411479, −1.61461779332635110673074836096, 0,
1.61461779332635110673074836096, 2.28956610059431075582349411479, 3.62406434056639048963936774853, 4.23894776027283377880109172566, 5.13608091937105447697975783959, 5.91892391219779025956042628760, 6.60083796180378457282385209064, 7.59256477164320569305091666261, 8.097096554375915957754317563274