| L(s) = 1 | − 1.76·3-s − 1.36·5-s + 0.174·7-s + 0.119·9-s + 1.83·11-s + 3.53·13-s + 2.41·15-s − 6.42·17-s − 19-s − 0.307·21-s + 4.41·23-s − 3.12·25-s + 5.08·27-s + 0.386·29-s − 6.29·31-s − 3.24·33-s − 0.238·35-s + 7.89·37-s − 6.24·39-s − 1.09·41-s + 5.58·43-s − 0.163·45-s + 8.67·47-s − 6.96·49-s + 11.3·51-s + 53-s − 2.51·55-s + ⋯ |
| L(s) = 1 | − 1.01·3-s − 0.612·5-s + 0.0658·7-s + 0.0397·9-s + 0.553·11-s + 0.981·13-s + 0.624·15-s − 1.55·17-s − 0.229·19-s − 0.0671·21-s + 0.920·23-s − 0.624·25-s + 0.979·27-s + 0.0717·29-s − 1.13·31-s − 0.564·33-s − 0.0403·35-s + 1.29·37-s − 1.00·39-s − 0.171·41-s + 0.851·43-s − 0.0243·45-s + 1.26·47-s − 0.995·49-s + 1.58·51-s + 0.137·53-s − 0.339·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 + 1.76T + 3T^{2} \) |
| 5 | \( 1 + 1.36T + 5T^{2} \) |
| 7 | \( 1 - 0.174T + 7T^{2} \) |
| 11 | \( 1 - 1.83T + 11T^{2} \) |
| 13 | \( 1 - 3.53T + 13T^{2} \) |
| 17 | \( 1 + 6.42T + 17T^{2} \) |
| 23 | \( 1 - 4.41T + 23T^{2} \) |
| 29 | \( 1 - 0.386T + 29T^{2} \) |
| 31 | \( 1 + 6.29T + 31T^{2} \) |
| 37 | \( 1 - 7.89T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 - 5.58T + 43T^{2} \) |
| 47 | \( 1 - 8.67T + 47T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 2.20T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−8.055163078380566889982145466165, −7.22966799732396108724236990115, −6.42867122526090328845104889272, −6.02203147196923295848586851866, −5.07746882310225651320134025419, −4.30371841325542575064410386184, −3.64488273149336947518520296212, −2.43438061178088378319407695413, −1.14235944175899039301871146016, 0,
1.14235944175899039301871146016, 2.43438061178088378319407695413, 3.64488273149336947518520296212, 4.30371841325542575064410386184, 5.07746882310225651320134025419, 6.02203147196923295848586851866, 6.42867122526090328845104889272, 7.22966799732396108724236990115, 8.055163078380566889982145466165
