| L(s) = 1 | + 2.58·2-s − 3-s + 4.65·4-s − 2.58·6-s − 1.98·7-s + 6.85·8-s + 9-s − 2.95·11-s − 4.65·12-s + 0.946·13-s − 5.12·14-s + 8.37·16-s + 0.496·17-s + 2.58·18-s − 4.07·19-s + 1.98·21-s − 7.63·22-s − 3.49·23-s − 6.85·24-s + 2.44·26-s − 27-s − 9.24·28-s − 4.14·29-s + 3.32·31-s + 7.89·32-s + 2.95·33-s + 1.28·34-s + ⋯ |
| L(s) = 1 | + 1.82·2-s − 0.577·3-s + 2.32·4-s − 1.05·6-s − 0.750·7-s + 2.42·8-s + 0.333·9-s − 0.891·11-s − 1.34·12-s + 0.262·13-s − 1.36·14-s + 2.09·16-s + 0.120·17-s + 0.608·18-s − 0.934·19-s + 0.433·21-s − 1.62·22-s − 0.729·23-s − 1.39·24-s + 0.478·26-s − 0.192·27-s − 1.74·28-s − 0.770·29-s + 0.598·31-s + 1.39·32-s + 0.514·33-s + 0.219·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 107 | \( 1 - T \) |
| good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 7 | \( 1 + 1.98T + 7T^{2} \) |
| 11 | \( 1 + 2.95T + 11T^{2} \) |
| 13 | \( 1 - 0.946T + 13T^{2} \) |
| 17 | \( 1 - 0.496T + 17T^{2} \) |
| 19 | \( 1 + 4.07T + 19T^{2} \) |
| 23 | \( 1 + 3.49T + 23T^{2} \) |
| 29 | \( 1 + 4.14T + 29T^{2} \) |
| 31 | \( 1 - 3.32T + 31T^{2} \) |
| 37 | \( 1 - 4.53T + 37T^{2} \) |
| 41 | \( 1 + 4.76T + 41T^{2} \) |
| 43 | \( 1 + 5.56T + 43T^{2} \) |
| 47 | \( 1 + 0.869T + 47T^{2} \) |
| 53 | \( 1 - 6.42T + 53T^{2} \) |
| 59 | \( 1 + 1.71T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 2.08T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 6.82T + 97T^{2} \) |
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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
−7.03962076370941901933982373599, −6.57298031805760636142230313350, −5.80993517400710205847251511436, −5.54132129091736462496107639373, −4.58587191648106980637016543295, −4.10296468926343050204179555630, −3.25629615407396090886232918244, −2.58362077512136322760776692825, −1.66117356543528403726866762355, 0,
1.66117356543528403726866762355, 2.58362077512136322760776692825, 3.25629615407396090886232918244, 4.10296468926343050204179555630, 4.58587191648106980637016543295, 5.54132129091736462496107639373, 5.80993517400710205847251511436, 6.57298031805760636142230313350, 7.03962076370941901933982373599
