L(s) = 1 | − 20.9·2-s − 81·3-s − 72.2·4-s + 1.69e3·6-s − 3.57e3·7-s + 1.22e4·8-s + 6.56e3·9-s − 7.45e4·11-s + 5.85e3·12-s + 3.44e4·13-s + 7.49e4·14-s − 2.19e5·16-s − 3.72e5·17-s − 1.37e5·18-s − 8.35e5·19-s + 2.89e5·21-s + 1.56e6·22-s − 3.18e4·23-s − 9.92e5·24-s − 7.23e5·26-s − 5.31e5·27-s + 2.58e5·28-s − 6.73e6·29-s − 4.04e6·31-s − 1.66e6·32-s + 6.03e6·33-s + 7.81e6·34-s + ⋯ |
L(s) = 1 | − 0.926·2-s − 0.577·3-s − 0.141·4-s + 0.535·6-s − 0.562·7-s + 1.05·8-s + 0.333·9-s − 1.53·11-s + 0.0814·12-s + 0.334·13-s + 0.521·14-s − 0.838·16-s − 1.08·17-s − 0.308·18-s − 1.47·19-s + 0.324·21-s + 1.42·22-s − 0.0237·23-s − 0.610·24-s − 0.310·26-s − 0.192·27-s + 0.0794·28-s − 1.76·29-s − 0.786·31-s − 0.280·32-s + 0.885·33-s + 1.00·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.2331883349\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2331883349\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 20.9T + 512T^{2} \) |
| 7 | \( 1 + 3.57e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.45e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.44e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.72e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.35e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.18e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.73e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.04e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.29e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.10e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.42e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.76e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.30e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.04e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.99e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.98e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 4.52e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.64e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.16e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.77e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.06e9T + 7.60e17T^{2} \) |
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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
−12.92857653356707930978745724555, −11.07701451076380340644196060686, −10.45685733013767206333911372205, −9.310172448212495034150435380685, −8.216282383744351610890139761452, −7.01684826126683809595640566491, −5.56065762722815345027300034248, −4.17716236772983254094269847577, −2.12270992369312587219717857991, −0.32478098279394818568167629543,
0.32478098279394818568167629543, 2.12270992369312587219717857991, 4.17716236772983254094269847577, 5.56065762722815345027300034248, 7.01684826126683809595640566491, 8.216282383744351610890139761452, 9.310172448212495034150435380685, 10.45685733013767206333911372205, 11.07701451076380340644196060686, 12.92857653356707930978745724555