| L(s) = 1 | − 0.797·3-s − 2.84·5-s − 0.0983·7-s − 2.36·9-s + 1.03·11-s − 3.18·13-s + 2.26·15-s − 17-s − 6.45·19-s + 0.0783·21-s + 4.89·23-s + 3.09·25-s + 4.27·27-s − 1.16·29-s − 8.98·31-s − 0.822·33-s + 0.279·35-s − 6.37·37-s + 2.53·39-s + 4.73·41-s − 11.2·43-s + 6.72·45-s − 10.9·47-s − 6.99·49-s + 0.797·51-s + 4.31·53-s − 2.93·55-s + ⋯ |
| L(s) = 1 | − 0.460·3-s − 1.27·5-s − 0.0371·7-s − 0.788·9-s + 0.311·11-s − 0.882·13-s + 0.585·15-s − 0.242·17-s − 1.48·19-s + 0.0171·21-s + 1.02·23-s + 0.618·25-s + 0.823·27-s − 0.217·29-s − 1.61·31-s − 0.143·33-s + 0.0472·35-s − 1.04·37-s + 0.406·39-s + 0.738·41-s − 1.71·43-s + 1.00·45-s − 1.59·47-s − 0.998·49-s + 0.111·51-s + 0.593·53-s − 0.395·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1823687419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1823687419\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 0.797T + 3T^{2} \) |
| 5 | \( 1 + 2.84T + 5T^{2} \) |
| 7 | \( 1 + 0.0983T + 7T^{2} \) |
| 11 | \( 1 - 1.03T + 11T^{2} \) |
| 13 | \( 1 + 3.18T + 13T^{2} \) |
| 19 | \( 1 + 6.45T + 19T^{2} \) |
| 23 | \( 1 - 4.89T + 23T^{2} \) |
| 29 | \( 1 + 1.16T + 29T^{2} \) |
| 31 | \( 1 + 8.98T + 31T^{2} \) |
| 37 | \( 1 + 6.37T + 37T^{2} \) |
| 41 | \( 1 - 4.73T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 4.31T + 53T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 - 1.67T + 67T^{2} \) |
| 71 | \( 1 + 8.89T + 71T^{2} \) |
| 73 | \( 1 + 2.54T + 73T^{2} \) |
| 79 | \( 1 + 8.31T + 79T^{2} \) |
| 83 | \( 1 + 4.78T + 83T^{2} \) |
| 89 | \( 1 + 18.8T + 89T^{2} \) |
| 97 | \( 1 - 9.43T + 97T^{2} \) |
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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
−7.88665691134557412186722970227, −6.90840708292678362484755075785, −6.77032643261866354751759852612, −5.63207969878438846353849332506, −5.03297993338716594317056494881, −4.29815835265241589935328650965, −3.57526965939542986978424598523, −2.80825266013189132515312426846, −1.72989331560545318520690371112, −0.20763982621402241200641321876,
0.20763982621402241200641321876, 1.72989331560545318520690371112, 2.80825266013189132515312426846, 3.57526965939542986978424598523, 4.29815835265241589935328650965, 5.03297993338716594317056494881, 5.63207969878438846353849332506, 6.77032643261866354751759852612, 6.90840708292678362484755075785, 7.88665691134557412186722970227
