L(s) = 1 | + 2-s + 3-s + 4-s + 4.26·5-s + 6-s + 8-s + 9-s + 4.26·10-s − 3.76·11-s + 12-s − 2.26·13-s + 4.26·15-s + 16-s + 0.352·17-s + 18-s + 2·19-s + 4.26·20-s − 3.76·22-s − 23-s + 24-s + 13.1·25-s − 2.26·26-s + 27-s + 9.13·29-s + 4.26·30-s − 7.89·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.90·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.34·10-s − 1.13·11-s + 0.288·12-s − 0.626·13-s + 1.10·15-s + 0.250·16-s + 0.0855·17-s + 0.235·18-s + 0.458·19-s + 0.952·20-s − 0.803·22-s − 0.208·23-s + 0.204·24-s + 2.63·25-s − 0.443·26-s + 0.192·27-s + 1.69·29-s + 0.777·30-s − 1.41·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.842474605\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.842474605\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 4.26T + 5T^{2} \) |
| 11 | \( 1 + 3.76T + 11T^{2} \) |
| 13 | \( 1 + 2.26T + 13T^{2} \) |
| 17 | \( 1 - 0.352T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 29 | \( 1 - 9.13T + 29T^{2} \) |
| 31 | \( 1 + 7.89T + 31T^{2} \) |
| 37 | \( 1 - 3.15T + 37T^{2} \) |
| 41 | \( 1 - 5.90T + 41T^{2} \) |
| 43 | \( 1 - 9.67T + 43T^{2} \) |
| 47 | \( 1 - 9.27T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 9.74T + 59T^{2} \) |
| 61 | \( 1 - 1.12T + 61T^{2} \) |
| 67 | \( 1 + 8.99T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 - 3.01T + 73T^{2} \) |
| 79 | \( 1 - 3.01T + 79T^{2} \) |
| 83 | \( 1 - 7.22T + 83T^{2} \) |
| 89 | \( 1 - 4.35T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−7.70585514374195620698555968945, −7.35803575515117109999300131080, −6.24143613901803868735614955346, −5.93667898392993286679625931622, −5.05620039834147318610565943564, −4.67040245985694514077494029664, −3.38118957467883120114702822282, −2.50622236186389871823298359303, −2.29358038621653405491030038926, −1.14518776653151587738100845789,
1.14518776653151587738100845789, 2.29358038621653405491030038926, 2.50622236186389871823298359303, 3.38118957467883120114702822282, 4.67040245985694514077494029664, 5.05620039834147318610565943564, 5.93667898392993286679625931622, 6.24143613901803868735614955346, 7.35803575515117109999300131080, 7.70585514374195620698555968945