L(s) = 1 | − 2·2-s + 4·4-s − 14·7-s − 8·8-s − 3·11-s − 47·13-s + 28·14-s + 16·16-s − 39·17-s + 32·19-s + 6·22-s − 99·23-s + 94·26-s − 56·28-s − 51·29-s + 83·31-s − 32·32-s + 78·34-s − 314·37-s − 64·38-s + 108·41-s − 299·43-s − 12·44-s + 198·46-s + 531·47-s − 147·49-s − 188·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.0822·11-s − 1.00·13-s + 0.534·14-s + 1/4·16-s − 0.556·17-s + 0.386·19-s + 0.0581·22-s − 0.897·23-s + 0.709·26-s − 0.377·28-s − 0.326·29-s + 0.480·31-s − 0.176·32-s + 0.393·34-s − 1.39·37-s − 0.273·38-s + 0.411·41-s − 1.06·43-s − 0.0411·44-s + 0.634·46-s + 1.64·47-s − 3/7·49-s − 0.501·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7631366940\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7631366940\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 3 T + p^{3} T^{2} \) |
| 13 | \( 1 + 47 T + p^{3} T^{2} \) |
| 17 | \( 1 + 39 T + p^{3} T^{2} \) |
| 19 | \( 1 - 32 T + p^{3} T^{2} \) |
| 23 | \( 1 + 99 T + p^{3} T^{2} \) |
| 29 | \( 1 + 51 T + p^{3} T^{2} \) |
| 31 | \( 1 - 83 T + p^{3} T^{2} \) |
| 37 | \( 1 + 314 T + p^{3} T^{2} \) |
| 41 | \( 1 - 108 T + p^{3} T^{2} \) |
| 43 | \( 1 + 299 T + p^{3} T^{2} \) |
| 47 | \( 1 - 531 T + p^{3} T^{2} \) |
| 53 | \( 1 - 564 T + p^{3} T^{2} \) |
| 59 | \( 1 + 12 T + p^{3} T^{2} \) |
| 61 | \( 1 - 230 T + p^{3} T^{2} \) |
| 67 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 71 | \( 1 + 120 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1106 T + p^{3} T^{2} \) |
| 79 | \( 1 + 739 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1086 T + p^{3} T^{2} \) |
| 89 | \( 1 - 120 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1642 T + p^{3} T^{2} \) |
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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
−9.242600091318647644074089138311, −8.566159084959410809212431691149, −7.56509285133778011812258591471, −6.96185619498313739765028299876, −6.09887789655169037927699316238, −5.15088437590206611910291656701, −3.96132193738706139004392196255, −2.87834104011293302782275561667, −1.94446813425265686612180979521, −0.46415502122991868254312454540,
0.46415502122991868254312454540, 1.94446813425265686612180979521, 2.87834104011293302782275561667, 3.96132193738706139004392196255, 5.15088437590206611910291656701, 6.09887789655169037927699316238, 6.96185619498313739765028299876, 7.56509285133778011812258591471, 8.566159084959410809212431691149, 9.242600091318647644074089138311