L(s) = 1 | + 2·3-s − 3·7-s + 9-s − 5·11-s + 4·13-s + 3·17-s + 19-s − 6·21-s + 8·23-s − 4·27-s − 2·29-s − 4·31-s − 10·33-s − 10·37-s + 8·39-s + 10·41-s + 43-s − 47-s + 2·49-s + 6·51-s + 4·53-s + 2·57-s − 6·59-s − 13·61-s − 3·63-s − 12·67-s + 16·69-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 1.13·7-s + 1/3·9-s − 1.50·11-s + 1.10·13-s + 0.727·17-s + 0.229·19-s − 1.30·21-s + 1.66·23-s − 0.769·27-s − 0.371·29-s − 0.718·31-s − 1.74·33-s − 1.64·37-s + 1.28·39-s + 1.56·41-s + 0.152·43-s − 0.145·47-s + 2/7·49-s + 0.840·51-s + 0.549·53-s + 0.264·57-s − 0.781·59-s − 1.66·61-s − 0.377·63-s − 1.46·67-s + 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 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
−7.47919328720127535476392340605, −7.17493337021089154068078801135, −5.99393029668735597586094848454, −5.60898136494309753729619314408, −4.60987047041952044980776106190, −3.46818377937813709652229576872, −3.19811739429940381844565992426, −2.56342586766102692928538950631, −1.40280534720434516884077987832, 0,
1.40280534720434516884077987832, 2.56342586766102692928538950631, 3.19811739429940381844565992426, 3.46818377937813709652229576872, 4.60987047041952044980776106190, 5.60898136494309753729619314408, 5.99393029668735597586094848454, 7.17493337021089154068078801135, 7.47919328720127535476392340605