L(s) = 1 | + 3·3-s − 18·5-s + 8·7-s + 9·9-s + 36·11-s − 10·13-s − 54·15-s + 18·17-s − 100·19-s + 24·21-s + 72·23-s + 199·25-s + 27·27-s − 234·29-s − 16·31-s + 108·33-s − 144·35-s − 226·37-s − 30·39-s + 90·41-s + 452·43-s − 162·45-s + 432·47-s − 279·49-s + 54·51-s + 414·53-s − 648·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.60·5-s + 0.431·7-s + 1/3·9-s + 0.986·11-s − 0.213·13-s − 0.929·15-s + 0.256·17-s − 1.20·19-s + 0.249·21-s + 0.652·23-s + 1.59·25-s + 0.192·27-s − 1.49·29-s − 0.0926·31-s + 0.569·33-s − 0.695·35-s − 1.00·37-s − 0.123·39-s + 0.342·41-s + 1.60·43-s − 0.536·45-s + 1.34·47-s − 0.813·49-s + 0.148·51-s + 1.07·53-s − 1.58·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9344401381\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9344401381\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
good | 5 | \( 1 + 18 T + p^{3} T^{2} \) |
| 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 234 T + p^{3} T^{2} \) |
| 31 | \( 1 + 16 T + p^{3} T^{2} \) |
| 37 | \( 1 + 226 T + p^{3} T^{2} \) |
| 41 | \( 1 - 90 T + p^{3} T^{2} \) |
| 43 | \( 1 - 452 T + p^{3} T^{2} \) |
| 47 | \( 1 - 432 T + p^{3} T^{2} \) |
| 53 | \( 1 - 414 T + p^{3} T^{2} \) |
| 59 | \( 1 + 684 T + p^{3} T^{2} \) |
| 61 | \( 1 - 422 T + p^{3} T^{2} \) |
| 67 | \( 1 - 332 T + p^{3} T^{2} \) |
| 71 | \( 1 + 360 T + p^{3} T^{2} \) |
| 73 | \( 1 - 26 T + p^{3} T^{2} \) |
| 79 | \( 1 - 512 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 630 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1054 T + p^{3} T^{2} \) |
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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
−19.65697221873960303781526402990, −18.91719553273081271326643124779, −16.94656258097691856593765242257, −15.42632448273370742373967743432, −14.49301964336732569099039232284, −12.46785072595897989496432329095, −11.13009747514994459636212243893, −8.800395059796186281786333535307, −7.38784077424888441275854326091, −4.02436353376270990090525228298,
4.02436353376270990090525228298, 7.38784077424888441275854326091, 8.800395059796186281786333535307, 11.13009747514994459636212243893, 12.46785072595897989496432329095, 14.49301964336732569099039232284, 15.42632448273370742373967743432, 16.94656258097691856593765242257, 18.91719553273081271326643124779, 19.65697221873960303781526402990