L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 6.10·11-s − 12-s − 1.68·13-s + 16-s + 6.83·17-s + 18-s − 1.68·19-s + 6.10·22-s + 2.31·23-s − 24-s − 1.68·26-s − 27-s + 8.41·29-s + 0.688·31-s + 32-s − 6.10·33-s + 6.83·34-s + 36-s + 2.31·37-s − 1.68·38-s + 1.68·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.84·11-s − 0.288·12-s − 0.468·13-s + 0.250·16-s + 1.65·17-s + 0.235·18-s − 0.387·19-s + 1.30·22-s + 0.481·23-s − 0.204·24-s − 0.331·26-s − 0.192·27-s + 1.56·29-s + 0.123·31-s + 0.176·32-s − 1.06·33-s + 1.17·34-s + 0.166·36-s + 0.380·37-s − 0.273·38-s + 0.270·39-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
\(L(1)\) |
\(\approx\) |
\(3.364853068\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.364853068\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 6.10T + 11T^{2} \) |
| 13 | \( 1 + 1.68T + 13T^{2} \) |
| 17 | \( 1 - 6.83T + 17T^{2} \) |
| 19 | \( 1 + 1.68T + 19T^{2} \) |
| 23 | \( 1 - 2.31T + 23T^{2} \) |
| 29 | \( 1 - 8.41T + 29T^{2} \) |
| 31 | \( 1 - 0.688T + 31T^{2} \) |
| 37 | \( 1 - 2.31T + 37T^{2} \) |
| 41 | \( 1 + 5.14T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 3.06T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 7.04T + 59T^{2} \) |
| 61 | \( 1 + 9.46T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 7.31T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 4.95T + 97T^{2} \) |
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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
−7.69610116099380479168524303134, −6.98843586894644509916755168547, −6.37857752476373876246153307315, −5.90448799699003769095891509909, −4.97626560075717723725936999739, −4.46469584591644050950193754134, −3.62039897116209834396287919849, −2.95380790334092450695975898964, −1.67308474202516400532378168033, −0.931381006030738259829183548305,
0.931381006030738259829183548305, 1.67308474202516400532378168033, 2.95380790334092450695975898964, 3.62039897116209834396287919849, 4.46469584591644050950193754134, 4.97626560075717723725936999739, 5.90448799699003769095891509909, 6.37857752476373876246153307315, 6.98843586894644509916755168547, 7.69610116099380479168524303134