L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 2·11-s + 16-s − 4·17-s + 6·19-s − 20-s − 2·22-s − 3·23-s + 25-s − 9·29-s + 4·31-s − 32-s + 4·34-s − 4·37-s − 6·38-s + 40-s − 7·41-s − 5·43-s + 2·44-s + 3·46-s + 8·47-s − 50-s + 2·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.603·11-s + 1/4·16-s − 0.970·17-s + 1.37·19-s − 0.223·20-s − 0.426·22-s − 0.625·23-s + 1/5·25-s − 1.67·29-s + 0.718·31-s − 0.176·32-s + 0.685·34-s − 0.657·37-s − 0.973·38-s + 0.158·40-s − 1.09·41-s − 0.762·43-s + 0.301·44-s + 0.442·46-s + 1.16·47-s − 0.141·50-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.065939793239333672640685356709, −7.26894921723600446928508659816, −6.81744244857954183150027466301, −5.89588864620231652033142375263, −5.09565695139434475827330528759, −4.05084036592834491429461078310, −3.37174455571566278352801880800, −2.27179767436923261006533101482, −1.28265228999086935733577332206, 0,
1.28265228999086935733577332206, 2.27179767436923261006533101482, 3.37174455571566278352801880800, 4.05084036592834491429461078310, 5.09565695139434475827330528759, 5.89588864620231652033142375263, 6.81744244857954183150027466301, 7.26894921723600446928508659816, 8.065939793239333672640685356709