| L(s) = 1 | − 2.23·3-s − 3.65·5-s + 7-s + 1.98·9-s + 3.59·11-s + 8.16·15-s + 0.523·17-s − 4.88·19-s − 2.23·21-s − 5.85·23-s + 8.35·25-s + 2.25·27-s − 0.335·29-s + 8.80·31-s − 8.02·33-s − 3.65·35-s + 5.06·37-s − 3.59·41-s + 9.42·43-s − 7.27·45-s − 11.0·47-s + 49-s − 1.16·51-s − 2.73·53-s − 13.1·55-s + 10.9·57-s + 10.7·59-s + ⋯ |
| L(s) = 1 | − 1.28·3-s − 1.63·5-s + 0.377·7-s + 0.663·9-s + 1.08·11-s + 2.10·15-s + 0.126·17-s − 1.12·19-s − 0.487·21-s − 1.22·23-s + 1.67·25-s + 0.434·27-s − 0.0623·29-s + 1.58·31-s − 1.39·33-s − 0.617·35-s + 0.832·37-s − 0.561·41-s + 1.43·43-s − 1.08·45-s − 1.61·47-s + 0.142·49-s − 0.163·51-s − 0.376·53-s − 1.77·55-s + 1.44·57-s + 1.39·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6087256098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6087256098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 + 3.65T + 5T^{2} \) |
| 11 | \( 1 - 3.59T + 11T^{2} \) |
| 17 | \( 1 - 0.523T + 17T^{2} \) |
| 19 | \( 1 + 4.88T + 19T^{2} \) |
| 23 | \( 1 + 5.85T + 23T^{2} \) |
| 29 | \( 1 + 0.335T + 29T^{2} \) |
| 31 | \( 1 - 8.80T + 31T^{2} \) |
| 37 | \( 1 - 5.06T + 37T^{2} \) |
| 41 | \( 1 + 3.59T + 41T^{2} \) |
| 43 | \( 1 - 9.42T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 2.73T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 0.0368T + 61T^{2} \) |
| 67 | \( 1 - 0.533T + 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 + 4.18T + 73T^{2} \) |
| 79 | \( 1 - 4.08T + 79T^{2} \) |
| 83 | \( 1 - 5.52T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−7.80134293371923091048086629245, −6.72776825463410321415225333769, −6.52238389829120121521752627241, −5.68964677825129586228288550944, −4.79760261779971837959424414632, −4.22272652790758174494898228671, −3.84220084552967957289456038839, −2.67460819015064565084586253119, −1.35296353635223372684775869176, −0.43792658762445472327521369949,
0.43792658762445472327521369949, 1.35296353635223372684775869176, 2.67460819015064565084586253119, 3.84220084552967957289456038839, 4.22272652790758174494898228671, 4.79760261779971837959424414632, 5.68964677825129586228288550944, 6.52238389829120121521752627241, 6.72776825463410321415225333769, 7.80134293371923091048086629245
