| L(s) = 1 | + 3-s + 3.83·5-s − 4.48·7-s + 9-s − 1.10·11-s − 5.67·13-s + 3.83·15-s − 0.581·17-s + 4.72·19-s − 4.48·21-s + 3.33·23-s + 9.72·25-s + 27-s + 3.38·29-s − 2.78·31-s − 1.10·33-s − 17.1·35-s + 4.43·37-s − 5.67·39-s + 5.90·41-s − 10.0·43-s + 3.83·45-s + 12.4·47-s + 13.0·49-s − 0.581·51-s + 5.10·53-s − 4.24·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.71·5-s − 1.69·7-s + 0.333·9-s − 0.333·11-s − 1.57·13-s + 0.990·15-s − 0.140·17-s + 1.08·19-s − 0.978·21-s + 0.695·23-s + 1.94·25-s + 0.192·27-s + 0.628·29-s − 0.499·31-s − 0.192·33-s − 2.90·35-s + 0.728·37-s − 0.908·39-s + 0.921·41-s − 1.52·43-s + 0.571·45-s + 1.80·47-s + 1.86·49-s − 0.0813·51-s + 0.701·53-s − 0.572·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.728013025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.728013025\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 3.83T + 5T^{2} \) |
| 7 | \( 1 + 4.48T + 7T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 + 5.67T + 13T^{2} \) |
| 17 | \( 1 + 0.581T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 23 | \( 1 - 3.33T + 23T^{2} \) |
| 29 | \( 1 - 3.38T + 29T^{2} \) |
| 31 | \( 1 + 2.78T + 31T^{2} \) |
| 37 | \( 1 - 4.43T + 37T^{2} \) |
| 41 | \( 1 - 5.90T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 5.10T + 53T^{2} \) |
| 59 | \( 1 - 2.77T + 59T^{2} \) |
| 61 | \( 1 - 9.42T + 61T^{2} \) |
| 67 | \( 1 + 9.07T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 9.14T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 + 1.50T + 89T^{2} \) |
| 97 | \( 1 - 0.761T + 97T^{2} \) |
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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
−7.66545906894652989091494023944, −7.00310536776741136957016519745, −6.58102193417165106262428402016, −5.66109831683722380487480967751, −5.29707907966295399733398889226, −4.28286542923298333044922799299, −3.07640677691538403117644255164, −2.75847918444987791047064168165, −2.05208593552907738799192087809, −0.77070989295029803958593360929,
0.77070989295029803958593360929, 2.05208593552907738799192087809, 2.75847918444987791047064168165, 3.07640677691538403117644255164, 4.28286542923298333044922799299, 5.29707907966295399733398889226, 5.66109831683722380487480967751, 6.58102193417165106262428402016, 7.00310536776741136957016519745, 7.66545906894652989091494023944
