L(s) = 1 | + 2·7-s + 6·13-s − 6·17-s + 2·19-s − 8·23-s − 5·25-s + 2·29-s − 4·31-s + 2·37-s − 10·41-s + 6·43-s + 4·47-s − 3·49-s − 4·53-s − 4·59-s + 2·61-s − 8·67-s − 12·71-s + 2·73-s − 14·79-s − 4·83-s + 12·91-s + 2·97-s − 14·101-s − 16·103-s + 4·107-s − 10·109-s + ⋯ |
L(s) = 1 | + 0.755·7-s + 1.66·13-s − 1.45·17-s + 0.458·19-s − 1.66·23-s − 25-s + 0.371·29-s − 0.718·31-s + 0.328·37-s − 1.56·41-s + 0.914·43-s + 0.583·47-s − 3/7·49-s − 0.549·53-s − 0.520·59-s + 0.256·61-s − 0.977·67-s − 1.42·71-s + 0.234·73-s − 1.57·79-s − 0.439·83-s + 1.25·91-s + 0.203·97-s − 1.39·101-s − 1.57·103-s + 0.386·107-s − 0.957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.53132753837424459565387809235, −6.65538524121335935736951890557, −6.01624298500087106098727240501, −5.48883417736729230439987281816, −4.38863218231925665898837742520, −4.06649425401559734263429239216, −3.11557930476375974767686690063, −1.99948079163893685970463261538, −1.43553598633874759690956925011, 0,
1.43553598633874759690956925011, 1.99948079163893685970463261538, 3.11557930476375974767686690063, 4.06649425401559734263429239216, 4.38863218231925665898837742520, 5.48883417736729230439987281816, 6.01624298500087106098727240501, 6.65538524121335935736951890557, 7.53132753837424459565387809235