L(s) = 1 | − 3·3-s + 2·7-s + 9·9-s − 22·13-s + 26·19-s − 6·21-s + 25·25-s − 27·27-s − 46·31-s + 26·37-s + 66·39-s − 22·43-s − 45·49-s − 78·57-s + 74·61-s + 18·63-s + 122·67-s − 46·73-s − 75·75-s − 142·79-s + 81·81-s − 44·91-s + 138·93-s + 2·97-s + 194·103-s − 214·109-s − 78·111-s + ⋯ |
L(s) = 1 | − 3-s + 2/7·7-s + 9-s − 1.69·13-s + 1.36·19-s − 2/7·21-s + 25-s − 27-s − 1.48·31-s + 0.702·37-s + 1.69·39-s − 0.511·43-s − 0.918·49-s − 1.36·57-s + 1.21·61-s + 2/7·63-s + 1.82·67-s − 0.630·73-s − 75-s − 1.79·79-s + 81-s − 0.483·91-s + 1.48·93-s + 2/97·97-s + 1.88·103-s − 1.96·109-s − 0.702·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5948837930\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5948837930\) |
\(L(2)\) |
|
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 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 - 2 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + 22 T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 - 26 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 + 46 T + p^{2} T^{2} \) |
| 37 | \( 1 - 26 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 22 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 74 T + p^{2} T^{2} \) |
| 67 | \( 1 - 122 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 46 T + p^{2} T^{2} \) |
| 79 | \( 1 + 142 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 2 T + p^{2} T^{2} \) |
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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
−20.13648884099051383753867625296, −18.53954984864775463547117835278, −17.38289166127045388071167095206, −16.25430910409664841934738329357, −14.63196342645636097137676549367, −12.68402344188096004229678663046, −11.42349289220851819736674535341, −9.811598721882561483753931498812, −7.25719115995708020350770369495, −5.13182975921542995889351136788,
5.13182975921542995889351136788, 7.25719115995708020350770369495, 9.811598721882561483753931498812, 11.42349289220851819736674535341, 12.68402344188096004229678663046, 14.63196342645636097137676549367, 16.25430910409664841934738329357, 17.38289166127045388071167095206, 18.53954984864775463547117835278, 20.13648884099051383753867625296