L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 4·11-s + 13-s − 15-s − 6·19-s − 2·21-s − 6·23-s + 25-s − 27-s − 8·29-s − 4·33-s + 2·35-s + 10·37-s − 39-s − 10·41-s − 12·43-s + 45-s − 12·47-s − 3·49-s − 6·53-s + 4·55-s + 6·57-s + 12·59-s − 10·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.258·15-s − 1.37·19-s − 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s − 0.696·33-s + 0.338·35-s + 1.64·37-s − 0.160·39-s − 1.56·41-s − 1.82·43-s + 0.149·45-s − 1.75·47-s − 3/7·49-s − 0.824·53-s + 0.539·55-s + 0.794·57-s + 1.56·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.76227055065499443350355283896, −6.71745424614644641200331157103, −6.32885196070129950389736854793, −5.66166404993959955330612709932, −4.75520568730172280777787452009, −4.19862029073904270457437696069, −3.34273506001367892274300025321, −1.88743845190069510184966974536, −1.56171986626511646412430492713, 0,
1.56171986626511646412430492713, 1.88743845190069510184966974536, 3.34273506001367892274300025321, 4.19862029073904270457437696069, 4.75520568730172280777787452009, 5.66166404993959955330612709932, 6.32885196070129950389736854793, 6.71745424614644641200331157103, 7.76227055065499443350355283896