L(s) = 1 | + 2-s − 0.906·3-s + 4-s − 1.94·5-s − 0.906·6-s − 7-s + 8-s − 2.17·9-s − 1.94·10-s − 2.64·11-s − 0.906·12-s − 3.95·13-s − 14-s + 1.76·15-s + 16-s − 1.48·17-s − 2.17·18-s − 1.94·20-s + 0.906·21-s − 2.64·22-s − 5.29·23-s − 0.906·24-s − 1.21·25-s − 3.95·26-s + 4.69·27-s − 28-s + 9.62·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.523·3-s + 0.5·4-s − 0.870·5-s − 0.370·6-s − 0.377·7-s + 0.353·8-s − 0.725·9-s − 0.615·10-s − 0.796·11-s − 0.261·12-s − 1.09·13-s − 0.267·14-s + 0.455·15-s + 0.250·16-s − 0.359·17-s − 0.513·18-s − 0.435·20-s + 0.197·21-s − 0.563·22-s − 1.10·23-s − 0.185·24-s − 0.242·25-s − 0.775·26-s + 0.903·27-s − 0.188·28-s + 1.78·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9207427880\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9207427880\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 0.906T + 3T^{2} \) |
| 5 | \( 1 + 1.94T + 5T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 + 3.95T + 13T^{2} \) |
| 17 | \( 1 + 1.48T + 17T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 - 9.62T + 29T^{2} \) |
| 31 | \( 1 + 4.31T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 4.90T + 43T^{2} \) |
| 47 | \( 1 - 5.28T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 3.51T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 2.65T + 67T^{2} \) |
| 71 | \( 1 - 9.09T + 71T^{2} \) |
| 73 | \( 1 + 6.52T + 73T^{2} \) |
| 79 | \( 1 - 0.0414T + 79T^{2} \) |
| 83 | \( 1 + 5.98T + 83T^{2} \) |
| 89 | \( 1 - 9.22T + 89T^{2} \) |
| 97 | \( 1 - 1.40T + 97T^{2} \) |
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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
−8.024259262781067044618420187771, −7.46424706680092967035530432335, −6.70881370860433473567708721696, −5.94172739771695779993885400699, −5.28859754642028551816998668528, −4.59532768085161058371062616970, −3.83812779384524445799315455168, −2.90798605511056167624099523340, −2.24236914791131952481181042588, −0.44815842703445785048263945943,
0.44815842703445785048263945943, 2.24236914791131952481181042588, 2.90798605511056167624099523340, 3.83812779384524445799315455168, 4.59532768085161058371062616970, 5.28859754642028551816998668528, 5.94172739771695779993885400699, 6.70881370860433473567708721696, 7.46424706680092967035530432335, 8.024259262781067044618420187771