L(s) = 1 | + 3-s + 5-s − 2·9-s − 2·11-s − 5·13-s + 15-s − 4·17-s + 2·19-s + 23-s + 25-s − 5·27-s − 3·29-s − 7·31-s − 2·33-s − 2·37-s − 5·39-s − 9·41-s + 4·43-s − 2·45-s + 9·47-s − 7·49-s − 4·51-s − 6·53-s − 2·55-s + 2·57-s + 2·61-s − 5·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s − 0.603·11-s − 1.38·13-s + 0.258·15-s − 0.970·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s − 0.962·27-s − 0.557·29-s − 1.25·31-s − 0.348·33-s − 0.328·37-s − 0.800·39-s − 1.40·41-s + 0.609·43-s − 0.298·45-s + 1.31·47-s − 49-s − 0.560·51-s − 0.824·53-s − 0.269·55-s + 0.264·57-s + 0.256·61-s − 0.620·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−9.059764867160173848395903262156, −8.042512496320936634605471860821, −7.41139264495264264730579840356, −6.54324019023144243014049017468, −5.45442377986721190421165029441, −4.93644916229499082757206892360, −3.65371875750472760129010692065, −2.67165325301574617098367487147, −1.99295234586096061072148262074, 0,
1.99295234586096061072148262074, 2.67165325301574617098367487147, 3.65371875750472760129010692065, 4.93644916229499082757206892360, 5.45442377986721190421165029441, 6.54324019023144243014049017468, 7.41139264495264264730579840356, 8.042512496320936634605471860821, 9.059764867160173848395903262156