| L(s) = 1 | − 0.201·2-s − 0.798·3-s − 1.95·4-s + 0.160·6-s − 1.75·7-s + 0.798·8-s − 2.36·9-s + 1.56·12-s − 2.79·13-s + 0.354·14-s + 3.75·16-s + 5.71·17-s + 0.476·18-s − 0.757·19-s + 1.40·21-s + 3.71·23-s − 0.637·24-s + 0.564·26-s + 4.28·27-s + 3.44·28-s + 6.95·29-s + 7.51·31-s − 2.35·32-s − 1.15·34-s + 4.62·36-s + 1.79·37-s + 0.152·38-s + ⋯ |
| L(s) = 1 | − 0.142·2-s − 0.460·3-s − 0.979·4-s + 0.0657·6-s − 0.664·7-s + 0.282·8-s − 0.787·9-s + 0.451·12-s − 0.776·13-s + 0.0947·14-s + 0.939·16-s + 1.38·17-s + 0.112·18-s − 0.173·19-s + 0.306·21-s + 0.775·23-s − 0.130·24-s + 0.110·26-s + 0.823·27-s + 0.650·28-s + 1.29·29-s + 1.34·31-s − 0.416·32-s − 0.197·34-s + 0.771·36-s + 0.295·37-s + 0.0247·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.201T + 2T^{2} \) |
| 3 | \( 1 + 0.798T + 3T^{2} \) |
| 7 | \( 1 + 1.75T + 7T^{2} \) |
| 13 | \( 1 + 2.79T + 13T^{2} \) |
| 17 | \( 1 - 5.71T + 17T^{2} \) |
| 19 | \( 1 + 0.757T + 19T^{2} \) |
| 23 | \( 1 - 3.71T + 23T^{2} \) |
| 29 | \( 1 - 6.95T + 29T^{2} \) |
| 31 | \( 1 - 7.51T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 41 | \( 1 - 4.51T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 8.20T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 1.04T + 59T^{2} \) |
| 61 | \( 1 + 7.20T + 61T^{2} \) |
| 67 | \( 1 + 0.362T + 67T^{2} \) |
| 71 | \( 1 - 2.76T + 71T^{2} \) |
| 73 | \( 1 - 8.47T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.254221977039117055699504669370, −7.87787458599553965344438313555, −6.66135165857437862782394306960, −6.09247907480869997669309557190, −5.05595944468701090649466744261, −4.77024877454006157550728520442, −3.42550432302856239171241615870, −2.85072526700374314313083988945, −1.10366014998179732918885785265, 0,
1.10366014998179732918885785265, 2.85072526700374314313083988945, 3.42550432302856239171241615870, 4.77024877454006157550728520442, 5.05595944468701090649466744261, 6.09247907480869997669309557190, 6.66135165857437862782394306960, 7.87787458599553965344438313555, 8.254221977039117055699504669370
