| L(s) = 1 | + 2-s + 2.74·3-s + 4-s + 1.89·5-s + 2.74·6-s + 2.43·7-s + 8-s + 4.53·9-s + 1.89·10-s − 4.99·11-s + 2.74·12-s − 1.80·13-s + 2.43·14-s + 5.20·15-s + 16-s + 1.87·17-s + 4.53·18-s − 7.01·19-s + 1.89·20-s + 6.68·21-s − 4.99·22-s + 23-s + 2.74·24-s − 1.41·25-s − 1.80·26-s + 4.20·27-s + 2.43·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.58·3-s + 0.5·4-s + 0.847·5-s + 1.12·6-s + 0.920·7-s + 0.353·8-s + 1.51·9-s + 0.599·10-s − 1.50·11-s + 0.792·12-s − 0.500·13-s + 0.650·14-s + 1.34·15-s + 0.250·16-s + 0.455·17-s + 1.06·18-s − 1.61·19-s + 0.423·20-s + 1.45·21-s − 1.06·22-s + 0.208·23-s + 0.560·24-s − 0.282·25-s − 0.353·26-s + 0.810·27-s + 0.460·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.947551493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.947551493\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.74T + 3T^{2} \) |
| 5 | \( 1 - 1.89T + 5T^{2} \) |
| 7 | \( 1 - 2.43T + 7T^{2} \) |
| 11 | \( 1 + 4.99T + 11T^{2} \) |
| 13 | \( 1 + 1.80T + 13T^{2} \) |
| 17 | \( 1 - 1.87T + 17T^{2} \) |
| 19 | \( 1 + 7.01T + 19T^{2} \) |
| 31 | \( 1 - 4.50T + 31T^{2} \) |
| 37 | \( 1 + 5.69T + 37T^{2} \) |
| 41 | \( 1 + 3.91T + 41T^{2} \) |
| 43 | \( 1 - 1.45T + 43T^{2} \) |
| 47 | \( 1 - 4.23T + 47T^{2} \) |
| 53 | \( 1 + 7.14T + 53T^{2} \) |
| 59 | \( 1 + 6.11T + 59T^{2} \) |
| 61 | \( 1 + 1.81T + 61T^{2} \) |
| 67 | \( 1 - 1.34T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 5.94T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 - 4.40T + 83T^{2} \) |
| 89 | \( 1 - 1.05T + 89T^{2} \) |
| 97 | \( 1 - 5.62T + 97T^{2} \) |
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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
−9.652942693981514286589800051465, −8.589688799213082221598807182982, −8.059316691816998378628506840005, −7.39938169433670036592539252616, −6.26336939808063976077514399542, −5.19086792359785504744630031861, −4.53172711958388557888880075134, −3.32608350969651380090376370879, −2.37995111271027038092236960792, −1.89942714688812303038230374742,
1.89942714688812303038230374742, 2.37995111271027038092236960792, 3.32608350969651380090376370879, 4.53172711958388557888880075134, 5.19086792359785504744630031861, 6.26336939808063976077514399542, 7.39938169433670036592539252616, 8.059316691816998378628506840005, 8.589688799213082221598807182982, 9.652942693981514286589800051465
