| L(s) = 1 | + 2.64·2-s + 3-s + 5.02·4-s − 2.43·5-s + 2.64·6-s − 1.45·7-s + 8.00·8-s + 9-s − 6.45·10-s + 1.93·11-s + 5.02·12-s + 2.18·13-s − 3.85·14-s − 2.43·15-s + 11.1·16-s − 4.68·17-s + 2.64·18-s − 7.39·19-s − 12.2·20-s − 1.45·21-s + 5.13·22-s + 7.56·23-s + 8.00·24-s + 0.941·25-s + 5.79·26-s + 27-s − 7.31·28-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 0.577·3-s + 2.51·4-s − 1.09·5-s + 1.08·6-s − 0.550·7-s + 2.83·8-s + 0.333·9-s − 2.04·10-s + 0.583·11-s + 1.44·12-s + 0.606·13-s − 1.03·14-s − 0.629·15-s + 2.79·16-s − 1.13·17-s + 0.624·18-s − 1.69·19-s − 2.73·20-s − 0.317·21-s + 1.09·22-s + 1.57·23-s + 1.63·24-s + 0.188·25-s + 1.13·26-s + 0.192·27-s − 1.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.034877111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.034877111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 149 | \( 1 + T \) |
| good | 2 | \( 1 - 2.64T + 2T^{2} \) |
| 5 | \( 1 + 2.43T + 5T^{2} \) |
| 7 | \( 1 + 1.45T + 7T^{2} \) |
| 11 | \( 1 - 1.93T + 11T^{2} \) |
| 13 | \( 1 - 2.18T + 13T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 19 | \( 1 + 7.39T + 19T^{2} \) |
| 23 | \( 1 - 7.56T + 23T^{2} \) |
| 29 | \( 1 + 2.26T + 29T^{2} \) |
| 31 | \( 1 + 7.58T + 31T^{2} \) |
| 37 | \( 1 + 6.02T + 37T^{2} \) |
| 41 | \( 1 - 7.44T + 41T^{2} \) |
| 43 | \( 1 + 1.42T + 43T^{2} \) |
| 47 | \( 1 + 5.79T + 47T^{2} \) |
| 53 | \( 1 + 0.996T + 53T^{2} \) |
| 59 | \( 1 - 7.03T + 59T^{2} \) |
| 61 | \( 1 - 9.38T + 61T^{2} \) |
| 67 | \( 1 - 8.52T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 8.63T + 79T^{2} \) |
| 83 | \( 1 - 5.19T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−11.13335031239063122875157118265, −10.94540561660772211534442842723, −9.186400632041352473679858431512, −8.145138382876183265972090712805, −6.91411482529179861718813334962, −6.51220436150550257483374325001, −5.05422002814860093922298182160, −3.95151298682879786606738789055, −3.58078957968214236032933938593, −2.20709469387848236683707504653,
2.20709469387848236683707504653, 3.58078957968214236032933938593, 3.95151298682879786606738789055, 5.05422002814860093922298182160, 6.51220436150550257483374325001, 6.91411482529179861718813334962, 8.145138382876183265972090712805, 9.186400632041352473679858431512, 10.94540561660772211534442842723, 11.13335031239063122875157118265
