| L(s) = 1 | + 0.840·2-s + 3-s − 1.29·4-s + 2.74·5-s + 0.840·6-s + 7-s − 2.76·8-s + 9-s + 2.30·10-s − 5.62·11-s − 1.29·12-s − 3.65·13-s + 0.840·14-s + 2.74·15-s + 0.257·16-s − 5.86·17-s + 0.840·18-s − 2.05·19-s − 3.55·20-s + 21-s − 4.72·22-s + 2.78·23-s − 2.76·24-s + 2.54·25-s − 3.07·26-s + 27-s − 1.29·28-s + ⋯ |
| L(s) = 1 | + 0.594·2-s + 0.577·3-s − 0.646·4-s + 1.22·5-s + 0.343·6-s + 0.377·7-s − 0.978·8-s + 0.333·9-s + 0.730·10-s − 1.69·11-s − 0.373·12-s − 1.01·13-s + 0.224·14-s + 0.709·15-s + 0.0644·16-s − 1.42·17-s + 0.198·18-s − 0.472·19-s − 0.794·20-s + 0.218·21-s − 1.00·22-s + 0.580·23-s − 0.565·24-s + 0.509·25-s − 0.602·26-s + 0.192·27-s − 0.244·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 0.840T + 2T^{2} \) |
| 5 | \( 1 - 2.74T + 5T^{2} \) |
| 11 | \( 1 + 5.62T + 11T^{2} \) |
| 13 | \( 1 + 3.65T + 13T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 + 2.05T + 19T^{2} \) |
| 23 | \( 1 - 2.78T + 23T^{2} \) |
| 29 | \( 1 + 4.83T + 29T^{2} \) |
| 31 | \( 1 + 6.87T + 31T^{2} \) |
| 37 | \( 1 + 8.71T + 37T^{2} \) |
| 41 | \( 1 + 4.09T + 41T^{2} \) |
| 43 | \( 1 + 2.48T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 - 4.92T + 53T^{2} \) |
| 59 | \( 1 + 9.41T + 59T^{2} \) |
| 61 | \( 1 - 4.96T + 61T^{2} \) |
| 67 | \( 1 - 4.57T + 67T^{2} \) |
| 71 | \( 1 - 2.13T + 71T^{2} \) |
| 73 | \( 1 - 9.59T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 5.06T + 83T^{2} \) |
| 89 | \( 1 - 2.73T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−8.796881824827830843365819778475, −7.70238267445897695309459818605, −6.96896386281210364013217363053, −5.85658754025261944095891012238, −5.19112249410786507410836165283, −4.75023222838737545033259919734, −3.63321525868860762612857970041, −2.50431225623514826276977388219, −2.04887033637200668197075951327, 0,
2.04887033637200668197075951327, 2.50431225623514826276977388219, 3.63321525868860762612857970041, 4.75023222838737545033259919734, 5.19112249410786507410836165283, 5.85658754025261944095891012238, 6.96896386281210364013217363053, 7.70238267445897695309459818605, 8.796881824827830843365819778475
