L(s) = 1 | + 3-s − 7-s + 9-s − 4·11-s + 2·13-s + 2·17-s + 2·19-s − 21-s + 6·23-s + 27-s + 6·29-s − 6·31-s − 4·33-s − 4·37-s + 2·39-s + 4·43-s − 4·47-s + 49-s + 2·51-s − 2·53-s + 2·57-s − 4·59-s − 2·61-s − 63-s + 12·67-s + 6·69-s + 8·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.458·19-s − 0.218·21-s + 1.25·23-s + 0.192·27-s + 1.11·29-s − 1.07·31-s − 0.696·33-s − 0.657·37-s + 0.320·39-s + 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.280·51-s − 0.274·53-s + 0.264·57-s − 0.520·59-s − 0.256·61-s − 0.125·63-s + 1.46·67-s + 0.722·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.349880836\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.349880836\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.889113980155965024809490698610, −7.14194671441900948617052501880, −6.58114196420703854826408230901, −5.56581592745473848098823246985, −5.12295603946537873412325899473, −4.18157525096284393702339628873, −3.24401926273144337519240808462, −2.88372905255860291793004970069, −1.83281625668969637672688659028, −0.73050880376752979923237106010,
0.73050880376752979923237106010, 1.83281625668969637672688659028, 2.88372905255860291793004970069, 3.24401926273144337519240808462, 4.18157525096284393702339628873, 5.12295603946537873412325899473, 5.56581592745473848098823246985, 6.58114196420703854826408230901, 7.14194671441900948617052501880, 7.889113980155965024809490698610