L(s) = 1 | + 2·3-s + 5-s + 9-s − 3·11-s − 13-s + 2·15-s − 6·17-s + 19-s − 9·23-s + 25-s − 4·27-s + 6·29-s − 8·31-s − 6·33-s − 7·37-s − 2·39-s + 3·41-s − 2·43-s + 45-s − 9·47-s − 12·51-s + 9·53-s − 3·55-s + 2·57-s + 8·61-s − 65-s − 8·67-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.904·11-s − 0.277·13-s + 0.516·15-s − 1.45·17-s + 0.229·19-s − 1.87·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 1.43·31-s − 1.04·33-s − 1.15·37-s − 0.320·39-s + 0.468·41-s − 0.304·43-s + 0.149·45-s − 1.31·47-s − 1.68·51-s + 1.23·53-s − 0.404·55-s + 0.264·57-s + 1.02·61-s − 0.124·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.242231260856283599425760355867, −7.52976413603637449430925002571, −6.74597553381940693737004131495, −5.86491180213591020399547602028, −5.06877214247949001055253868167, −4.14569430363276678411348894634, −3.29187271919457833434346316534, −2.36803079032999184232461420098, −1.91864715026152841741755384973, 0,
1.91864715026152841741755384973, 2.36803079032999184232461420098, 3.29187271919457833434346316534, 4.14569430363276678411348894634, 5.06877214247949001055253868167, 5.86491180213591020399547602028, 6.74597553381940693737004131495, 7.52976413603637449430925002571, 8.242231260856283599425760355867