L(s) = 1 | + 3-s − 2·4-s + 3.46·5-s + 9-s + 5.73·11-s − 2·12-s − 5.46·13-s + 3.46·15-s + 4·16-s − 5·17-s − 0.535·19-s − 6.92·20-s + 5.46·23-s + 6.99·25-s + 27-s − 5.19·29-s − 5.73·31-s + 5.73·33-s − 2·36-s − 37-s − 5.46·39-s + 41-s + 9.92·43-s − 11.4·44-s + 3.46·45-s + 11.9·47-s + 4·48-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s + 1.54·5-s + 0.333·9-s + 1.72·11-s − 0.577·12-s − 1.51·13-s + 0.894·15-s + 16-s − 1.21·17-s − 0.122·19-s − 1.54·20-s + 1.13·23-s + 1.39·25-s + 0.192·27-s − 0.964·29-s − 1.02·31-s + 0.997·33-s − 0.333·36-s − 0.164·37-s − 0.874·39-s + 0.156·41-s + 1.51·43-s − 1.72·44-s + 0.516·45-s + 1.73·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.792789615\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.792789615\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 11 | \( 1 - 5.73T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 + 0.535T + 19T^{2} \) |
| 23 | \( 1 - 5.46T + 23T^{2} \) |
| 29 | \( 1 + 5.19T + 29T^{2} \) |
| 31 | \( 1 + 5.73T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 43 | \( 1 - 9.92T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 1.73T + 61T^{2} \) |
| 67 | \( 1 - 7.46T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 9.19T + 73T^{2} \) |
| 79 | \( 1 - 0.928T + 79T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 + 8.92T + 89T^{2} \) |
| 97 | \( 1 + 2.39T + 97T^{2} \) |
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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
−8.353678608770662847299865380460, −7.09519376516687918696654176803, −6.89885473750085340446559748583, −5.75050050996984274339697217848, −5.28906681727335924511498607093, −4.34428163346596125717697206755, −3.82311169850449025043110520212, −2.57498038315691314110759528709, −1.96474480712990866720892942998, −0.887454135928322001658700104375,
0.887454135928322001658700104375, 1.96474480712990866720892942998, 2.57498038315691314110759528709, 3.82311169850449025043110520212, 4.34428163346596125717697206755, 5.28906681727335924511498607093, 5.75050050996984274339697217848, 6.89885473750085340446559748583, 7.09519376516687918696654176803, 8.353678608770662847299865380460