L(s) = 1 | + 3-s − 7-s + 9-s − 4·11-s + 2·13-s − 2·17-s − 4·19-s − 21-s + 27-s − 2·29-s − 8·31-s − 4·33-s + 2·37-s + 2·39-s + 2·41-s + 4·43-s + 49-s − 2·51-s + 10·53-s − 4·57-s + 12·59-s + 6·61-s − 63-s + 12·67-s + 6·73-s + 4·77-s + 8·79-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 − 0.917·19-s − 0.218·21-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s + 0.328·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s + 1/7·49-s − 0.280·51-s + 1.37·53-s − 0.529·57-s + 1.56·59-s + 0.768·61-s − 0.125·63-s + 1.46·67-s + 0.702·73-s + 0.455·77-s + 0.900·79-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
\(L(1)\) |
\(\approx\) |
\(1.886500115\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.886500115\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−7.83126310022054818952447790457, −7.18227636268286237201187560988, −6.50153415479707969150943575899, −5.68574500195237832587977758841, −5.06097906025938069275471178732, −4.04963330296022757824072727958, −3.57164139731417709655196271519, −2.50847605028294654162945074975, −2.06366094226205196857658598380, −0.62589782358317546543571476764,
0.62589782358317546543571476764, 2.06366094226205196857658598380, 2.50847605028294654162945074975, 3.57164139731417709655196271519, 4.04963330296022757824072727958, 5.06097906025938069275471178732, 5.68574500195237832587977758841, 6.50153415479707969150943575899, 7.18227636268286237201187560988, 7.83126310022054818952447790457