L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 2·11-s + 12-s + 6·13-s + 16-s + 4·17-s − 18-s + 6·19-s − 2·22-s + 8·23-s − 24-s − 6·26-s + 27-s + 6·29-s + 2·31-s − 32-s + 2·33-s − 4·34-s + 36-s − 4·37-s − 6·38-s + 6·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s + 1.66·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.37·19-s − 0.426·22-s + 1.66·23-s − 0.204·24-s − 1.17·26-s + 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.176·32-s + 0.348·33-s − 0.685·34-s + 1/6·36-s − 0.657·37-s − 0.973·38-s + 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.607145739\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.607145739\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.093092979143138014933592632762, −7.21944480063252231791249372607, −6.79455148767677979831657599860, −5.90737019638486173594758717736, −5.21486636688170158529314536204, −4.12510422749134093310839877717, −3.29325978740021288883182836542, −2.85919145586828206199691202316, −1.37539524195221191352845435990, −1.08076206882020093348077554526,
1.08076206882020093348077554526, 1.37539524195221191352845435990, 2.85919145586828206199691202316, 3.29325978740021288883182836542, 4.12510422749134093310839877717, 5.21486636688170158529314536204, 5.90737019638486173594758717736, 6.79455148767677979831657599860, 7.21944480063252231791249372607, 8.093092979143138014933592632762