L(s) = 1 | − 2·3-s + 7-s + 9-s + 3·11-s − 4·13-s + 2·19-s − 2·21-s − 3·23-s + 4·27-s + 9·29-s + 8·31-s − 6·33-s + 5·37-s + 8·39-s − 6·41-s + 11·43-s + 6·47-s + 49-s + 6·53-s − 4·57-s − 10·61-s + 63-s + 5·67-s + 6·69-s + 15·71-s − 10·73-s + 3·77-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.904·11-s − 1.10·13-s + 0.458·19-s − 0.436·21-s − 0.625·23-s + 0.769·27-s + 1.67·29-s + 1.43·31-s − 1.04·33-s + 0.821·37-s + 1.28·39-s − 0.937·41-s + 1.67·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s − 0.529·57-s − 1.28·61-s + 0.125·63-s + 0.610·67-s + 0.722·69-s + 1.78·71-s − 1.17·73-s + 0.341·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.016943091\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.016943091\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−10.48340708300327334608204066535, −9.811649660309562675106236923024, −8.758337549295479378590072229628, −7.73562448419965250225099053149, −6.73034779603349979037904446310, −6.01118714823649537133179937027, −5.01328815310058208778255110773, −4.26504108499817032327826623435, −2.63651692432818486959856402458, −0.925578357219550016990910217681,
0.925578357219550016990910217681, 2.63651692432818486959856402458, 4.26504108499817032327826623435, 5.01328815310058208778255110773, 6.01118714823649537133179937027, 6.73034779603349979037904446310, 7.73562448419965250225099053149, 8.758337549295479378590072229628, 9.811649660309562675106236923024, 10.48340708300327334608204066535