| L(s) = 1 | − 2.17·2-s − 3-s + 2.73·4-s − 5-s + 2.17·6-s + 3.90·7-s − 1.59·8-s + 9-s + 2.17·10-s − 1.59·11-s − 2.73·12-s − 8.49·14-s + 15-s − 1.99·16-s − 3.76·17-s − 2.17·18-s + 7.91·19-s − 2.73·20-s − 3.90·21-s + 3.46·22-s + 6.22·23-s + 1.59·24-s + 25-s − 27-s + 10.6·28-s − 5.03·29-s − 2.17·30-s + ⋯ |
| L(s) = 1 | − 1.53·2-s − 0.577·3-s + 1.36·4-s − 0.447·5-s + 0.888·6-s + 1.47·7-s − 0.563·8-s + 0.333·9-s + 0.687·10-s − 0.480·11-s − 0.788·12-s − 2.27·14-s + 0.258·15-s − 0.499·16-s − 0.913·17-s − 0.512·18-s + 1.81·19-s − 0.610·20-s − 0.852·21-s + 0.738·22-s + 1.29·23-s + 0.325·24-s + 0.200·25-s − 0.192·27-s + 2.01·28-s − 0.935·29-s − 0.397·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7188088709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7188088709\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 7 | \( 1 - 3.90T + 7T^{2} \) |
| 11 | \( 1 + 1.59T + 11T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 - 7.91T + 19T^{2} \) |
| 23 | \( 1 - 6.22T + 23T^{2} \) |
| 29 | \( 1 + 5.03T + 29T^{2} \) |
| 31 | \( 1 + 0.184T + 31T^{2} \) |
| 37 | \( 1 - 1.64T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 6.74T + 43T^{2} \) |
| 47 | \( 1 - 6.58T + 47T^{2} \) |
| 53 | \( 1 + 5.51T + 53T^{2} \) |
| 59 | \( 1 + 4.32T + 59T^{2} \) |
| 61 | \( 1 + 9.73T + 61T^{2} \) |
| 67 | \( 1 + 6.16T + 67T^{2} \) |
| 71 | \( 1 + 1.24T + 71T^{2} \) |
| 73 | \( 1 + 2.25T + 73T^{2} \) |
| 79 | \( 1 + 4.33T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 + 9.14T + 89T^{2} \) |
| 97 | \( 1 - 9.58T + 97T^{2} \) |
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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
−9.082102469570517013370939922644, −8.036578039590484351792909087595, −7.53583446238671521750875384810, −7.15235409913216739969658551569, −5.87836195604451389003824930659, −4.99036963201235190797961490613, −4.32272438629174371921482316636, −2.77547586730896261100625640792, −1.61007759198707912042198494219, −0.74535747617501164678112605305,
0.74535747617501164678112605305, 1.61007759198707912042198494219, 2.77547586730896261100625640792, 4.32272438629174371921482316636, 4.99036963201235190797961490613, 5.87836195604451389003824930659, 7.15235409913216739969658551569, 7.53583446238671521750875384810, 8.036578039590484351792909087595, 9.082102469570517013370939922644
