| L(s) = 1 | + 2·3-s − 2·7-s + 9-s − 2·11-s − 2·13-s − 17-s − 4·21-s + 6·23-s − 4·27-s − 6·29-s + 10·31-s − 4·33-s − 2·37-s − 4·39-s + 10·41-s + 4·43-s + 12·47-s − 3·49-s − 2·51-s + 10·53-s − 8·59-s − 14·61-s − 2·63-s + 8·67-s + 12·69-s + 2·71-s + 14·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.242·17-s − 0.872·21-s + 1.25·23-s − 0.769·27-s − 1.11·29-s + 1.79·31-s − 0.696·33-s − 0.328·37-s − 0.640·39-s + 1.56·41-s + 0.609·43-s + 1.75·47-s − 3/7·49-s − 0.280·51-s + 1.37·53-s − 1.04·59-s − 1.79·61-s − 0.251·63-s + 0.977·67-s + 1.44·69-s + 0.237·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.384479316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.384479316\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−7.84695845550863160952620581646, −7.52889635490848705135417931621, −6.68213262689700777525173057493, −5.90962679434949949898139141800, −5.08071317327650874966709367089, −4.22016985038303405331172080823, −3.38944432830528237190605252191, −2.71554500700579199036677526073, −2.20985826491343379962521058827, −0.71815527987376246139952650178,
0.71815527987376246139952650178, 2.20985826491343379962521058827, 2.71554500700579199036677526073, 3.38944432830528237190605252191, 4.22016985038303405331172080823, 5.08071317327650874966709367089, 5.90962679434949949898139141800, 6.68213262689700777525173057493, 7.52889635490848705135417931621, 7.84695845550863160952620581646
