L(s) = 1 | + 3-s − 7-s + 9-s + 2·11-s + 2·13-s − 6·19-s − 21-s + 27-s − 6·29-s + 10·31-s + 2·33-s + 2·39-s + 6·41-s + 8·43-s + 12·47-s + 49-s + 6·53-s − 6·57-s − 6·61-s − 63-s − 4·67-s + 6·71-s + 14·73-s − 2·77-s + 4·79-s + 81-s − 6·87-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 1.37·19-s − 0.218·21-s + 0.192·27-s − 1.11·29-s + 1.79·31-s + 0.348·33-s + 0.320·39-s + 0.937·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 0.794·57-s − 0.768·61-s − 0.125·63-s − 0.488·67-s + 0.712·71-s + 1.63·73-s − 0.227·77-s + 0.450·79-s + 1/9·81-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.398326925\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.398326925\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.434245297150204148415288146074, −7.78639681133884465502736337734, −6.90387323042488557603119576349, −6.29327202941781976106993694381, −5.57555468896868896251857845353, −4.27630914070959476325236119988, −3.98209334202682588618811268971, −2.87902326798140581591211883883, −2.08648911117884467860480045356, −0.867994520545175267768641485821,
0.867994520545175267768641485821, 2.08648911117884467860480045356, 2.87902326798140581591211883883, 3.98209334202682588618811268971, 4.27630914070959476325236119988, 5.57555468896868896251857845353, 6.29327202941781976106993694381, 6.90387323042488557603119576349, 7.78639681133884465502736337734, 8.434245297150204148415288146074